#  CancerLLM: A Large Language Model in Cancer Domain

Mingchen Li1, Anne Blaes2, Steven Johnson3, Hongfang Liu4, Hua Xu5, Rui Zhang1
1Division of Computational Health Sciences, University of Minnesota Twin
Cities
2Division of Hematology, Oncology and Transplantation, University of Minnesota
Twin Cities
3Institute for Health Informatics, University of Minnesota Twin Cities
4McWilliams School of Biomedical Informatics, UTHealth Houston
5Department of Biomedical Informatics and Data Science, Yale School of
Medicine
1{li003378<EMAIL_ADDRESS>

###### Abstract

Medical Large Language Models (LLMs) such as ClinicalCamel 70B,
Llama3-OpenBioLLM 70B have demonstrated impressive performance on a wide
variety of medical NLP task. However, there still lacks a large language model
(LLM) specifically designed for cancer domain. Moreover, these LLMs typically
have billions of parameters, making them computationally expensive for
healthcare systems. Thus, in this study, we propose CancerLLM, a model with 7
billion parameters and a Mistral-style architecture, pre-trained on 2,676,642
clinical notes and 515,524 pathology reports covering 17 cancer types,
followed by fine-tuning on three cancer-relevant tasks, including cancer
phenotypes extraction, cancer diagnosis generation, and cancer treatment plan
generation. Our evaluation demonstrated that CancerLLM achieves state-of-the-
art results compared to other existing LLMs, with an average F1 score
improvement of 8.1%. Additionally, CancerLLM outperforms other models on two
proposed robustness testbeds. This illustrates that CancerLLM can be
effectively applied to clinical AI systems, enhancing clinical research and
healthcare delivery in the field of cancer.

##  1 Introduction

Recently, large language models (LLMs) like
GPT-4111https://openai.com/index/gpt-4/, and LLama2 (Touvron et al., 2023)
have become the dominant technology in various natural language processing
(NLP) tasks, these models develop impressive capabilities in different NLP
tasks, including information extraction (e.g., (Li & Zhang, 2023; Li et al.,
2024a; c; Zhang et al., 2024)), link prediction (e.g., (Li et al., 2024b; Liu
et al., 2024; Li et al., 2022)), and question answering (e.g., (Zhuang et al.,
2024; Jiang et al., 2024)). While much attention has been given to these
models’ capabilities in the general domain, it’s evident that specialist
models have the potential to significantly help clinical and biomedical
research Singhal et al. (2023). To tailor LLMs for the clinical domain,
several specialized models have been developed, such as ClinicalCamel 70B
(Toma et al., 2023) and Llama3-OpenBioLLM
70B222https://huggingface.co/aaditya/Llama3-OpenBioLLM-70B. Improving domain-
specific language models will accelerate breakthroughs in clinical and
biomedical research, leading to enhanced patient care.

Despite showcasing impressive general capabilities, current medical LLMs face
significant challenges when applied to cancer. This is primarily due to a
deficiency in cancer-specific knowledge within these models. This lack of
specialized understanding poses a barrier to the effective utilization of LLMs
in assisting doctors with cancer diagnosis and treatment planning. Expanding
cancer-specific knowledge within LLMs is imperative for surmounting these
obstacles and harnessing their full potential in the cancer field. By
integrating comprehensive information encompassing aspects like cancer
diagnosis, subjective information, objective information, nursing review of
systems (nursing ros) records, laboratory test results, patient self-
descriptions, medical history, treatment modalities, assessments, and patient
outcomes, LLMs can furnish healthcare professionals with more precise and
personalized recommendations. This augmented capability holds the promise of
revolutionizing cancer care and aiding medical professionals in enhancing
diagnostic accuracy and formulating treatment plans. To the best of our
knowledge, CancerBERT (Zhou et al., 2022) is the only model specifically
designed for the cancer domain. While CancerBERT (Zhou et al., 2022) stands
out as a specialized model for cancer-related tasks, its focus is primarily on
breast cancer, leaving other cancer types relatively unaddressed. This
limitation underscores the need for further development of specialized models
that cover a broader spectrum of cancer types and applications. By expanding
the scope of such models to encompass various cancer types and different
applications, healthcare professionals can access more comprehensive and
tailored resources to enhance patient care in the cancer domain.

Moreover, the deployment of large parameter LLMs presents a significant
challenge for hospitals or medical institutions with limited computational
resources. Models with a dozen billions of parameters, such as those models
with 13 billion parameters, like PMC LLaMA 13B (Wu et al., 2023) and Medalpaca
13B (Han et al., 2023), or those with 70 billion parameters, such as
Llama3-OpenBioLLM-70B333https://huggingface.co/aaditya/Llama3-OpenBioLLM-70B
and ClinicalCamel-70B (Toma et al., 2023), pose significant computational
challenges for hospitals or medical institutions with limited resources,
require substantial computing power and infrastructure to train and deploy
effectively. The financial and technical constraints associated with running
these models may pose barriers to their widespread adoption in cancer
settings. As such, there is a pressing need to develop a smaller LLM for the
cancer domain, reducing computational requirements to ensure that advanced
LLMs can benefit a broader range of medical professionals and patients.

Figure 1: The evolution of medical LLM performance on three tasks—cancer
phenotype extraction, diagnosis generation, and treatment plan generation—is
measured using the average F1 score, which includes Exact Match, BLEU-2, and
ROUGE-L. Our CancerLLM achieves the highest performance with an F1 score of
78.00%.

Bridging these gaps, in this work, we introduce CancerLLM, a state-of-the-art
LLM with 7 billion parameters designed to elevate LLM’s proficiency in the
cancer domain, and aid medical professionals in cancer phenotype extraction,
cancer diagnosis generation, and cancer treatment plan generation. Our model
represents an important step towards democratizing advancements in the cancer
domain of LLMs. More specifically, our research focuses on four steps: 1) pre-
training the LLM in the style of Mistral. We pre-train the Mistral 7B using
2,676,642 cancer clinical notes and 515,524 pathology reports from the
University of Minnesota (UMN) Clinical Data Repository. 2) fine-tuning the LLM
on a specialized dataset. We proposed three datasets focusing on phenotype
extraction, diagnosis generation and treatment plan generation. Following
this, we fine-tuned CancerLLM using instruction learning for the three
aforementioned tasks. 3) We evaluate the generation ability of CancerLLM
against other medical LLMs using Exact Match (Rajpurkar et al., 2016), BLEU-2
(Papineni et al., 2002), and ROUGE-L (Cohan & Goharian, 2016) metrics. 4) To
further evaluate the model’s ability, we propose two robustness testbeds which
include counterfactual robustness and misspellings robustness in the special
tasks. As shown in Figure 1, our model not only outperforms existing 7B and
13B models by a significant margin but also delivers good results with those
of 70B models.

The contributions of this work can be summarized as follows:

  * •

We proposed CancerLLM, a large language model which focuses on the cancer
domain.

  * •

An original study with the introduction of a benchmark of three tasks,
facilitating assessment against existing state-of-the-art open-source medical
LLMs.

  * •

We have developed three datasets aimed at fine-tuning CancerLLM for its three
downstream tasks.

  * •

A large in-depth quantitative analysis of the model’s generation ability and
robustness on proposed two testbeds.

##  2 CANCERLLM

Figure 2:  Overview of CancerLLM

Figure 2 illustrates the workflow of the CancerLLM, beginning with the
injection of cancer knowledge and followed by supervised instruction tuning.
Subsequently, we assess the generative capability of our framework compared to
current medical LLMs using three evaluation metrics. Additionally, we
introduce two testbeds to evaluate the robustness of the LLMs. Finally, the
trained CancerLLM is applied to three specific tasks.

###  2.1 Pre-training Data for Cancer

The data used in this study were obtained from the University of Minnesota
(UMN) Clinical Data Repository. It contains the health records of 31,465
cancer patients. Specifically, It includes 2,676,642 cancer clinical notes
(7.27GB) and 515,524 cancer pathology reports (536.85MB). We obtained the data
with the approval of the UMN Institutional Review Board (IRB). The statistics
information of patient and clinical notes are shown in Figure 3.

(a) Number of patients with different cancers (b) Number of clinical notes
with different cancers

Figure 3: Patients and clinical notes Statistics

###  2.2 Instruction Tuning Data for Cancer

To adapt CancerLLM for various downstream applications, we employ instruction
tuning for three distinct tasks: cancer phenotypes extraction, cancer
diagnosis generation, and cancer treatment plan generation. This section
provides detailed information on each task description and its instruction
tuning data.

####  2.2.1 Cancer Phenotypes extraction

Task description By providing sentences from cancer pathology reports or
cancer clinical notes, the model needs to extract eight specific entities:
hormone receptor type, hormone receptor status, tumor size, tumor site, cancer
grade, histological type, tumor laterality, and cancer stage.

Dataset The dataset (CancerNER) used in CancerBERT (Zhou et al., 2022) focuses
on named entity recognition (NER) of breast cancer-related phenotypes. This
data is extracted from both clinical notes and pathology reports within
electronic health records (EHRs). By inputting a sentence, CancerBERT is
expected to identify and recognize each entity corresponding to the
aforementioned types. LLMs have been found to underperform in NER tasks
compared to supervised models tailored for sequence labeling (Wang et al.,
2023). This is primarily due to the inherent differences between sequence
labeling tasks, which require the model to tag each token in a sequence, and
the generation tasks that LLMs are typically optimized for generating the next
tokens. So, in this paper, to enhance the performance of NER, we transformed
the CancerNER dataset into a question-answering format by posing eight
questions based on the entity types, the answers are relevant entities. The
question types are shown in Table 1. Note: by giving the input sentence
(context), when the question is not relevant with the context, our model is
expected to output "Not relevant". . Each instance follows the template
{instruction, context, response}. The instruction is: You are a medical
expert, this task involves answering the question based on the provided
context or text., The context is the sentence that needs to be processed,
along with a question. The response is the identified entity.

| question
---|---
1 | What is the tumor size in the given context?
2 | What is the histological type in the given context?
3 | Please identify the receptors mentioned in the provided context.
4 | What is the receptor type in the given context?
5 | Please identify the value of tumor laterality in the provided context.
6 | What is the stage of cancer in the given context?
7 | Please describe the tumor location in the given context
8 | What is the grade of cancer in the given context?

Table 1: Question types

####  2.2.2 Diagnosis Generation

Task description By giving the information from cancer clinical notes, which
includes the 1) reason for visit, 2) treatment site, 3) subjective
information, 4) nursing Review of Systems (ROS), 5) objective observations,
and 6) laboratory test results, the model is expected to generate the correct
cancer diagnosis.

Dataset We randomly selected 10,635 cancer clinical notes that do not overlap
with the pre-training dataset, and split them into 80% instances for training,
10% instances for testing, and 10% instances for validation. These datasets do
not overlap. Each instance follows the template {instruction, context,
response}. The instruction is: You are a medical expert. This task involves
generating the diagnosis based on the provided context or text, the context
includes the reason for visit, treatment site, subjective information, nursing
Review of Systems (ROS), objective observations, and laboratory test results.
The response is the cancer diagnosis.

####  2.2.3 Treatment Plan Generation

Task description By giving the information from clinical notes, which includes
the 1) reason for visit, 2) treatment site, 3) subjective information, 4)
nursing Review of Systems (ROS), 5) objective observations, 6) laboratory test
results, 7) diagnosis and 8) assessment, the model is expected to generate the
correct treatment plan.

Dataset We randomly selected 10,635 clinical notes that do not overlap with
the pre-training dataset, and split them into 80% instances for training, 10%
instances for testing, and 10% instances for validation. These datasets do not
overlap. Each instance follows the template {instruction, context, response}.
The instruction is: You are a medical expert. This task involves generating
the treatment plan based on the provided context or text, the context includes
the reason for visit, treatment site, subjective information, nursing Review
of Systems (ROS), objective observations, laboratory test results, diagnosis,
and assessment. The response is the treatment plan.

###  2.3 Training Methodology

Our training methods include two phases: continued pre-training and
instruction tuning. In this work, we choose Mistral 7B (Jiang et al., 2023) as
the foundation model to inject cancer knowledge in the pre-training progress,
as it has shown superior performance on various medical benchmarks. To improve
the efficiency of the training process, we utilize Low-Rank Adaptation (LoRA)
(Hu et al., 2021).

####  2.3.1 Continued Pre-training

This stage employs Mistral 7B, focusing on next-token prediction on clinical
notes and pathology reports. Due to computing resource limitations, we undergo
pre-training using LoRA. We parallelize the computation across ten 80G A100
GPUs, with each A100 utilizing around 40G of GPU memory. The total training
time is 47 hours. The training batch size and evaluation batch size per device
are set to 4. For LoRA, the rank is set to 8, alpha is set to 16, and dropout
is set to 0.05 to optimize learning. We use the AdamW optimizer and set the
learning rate to 2e-4. The training epoch is 1, and the evaluation step is
0.2.

####  2.3.2 Instruction Tuning

After continued pre-training, our model undergoes instruction tuning with
three datasets, aligning with different clinical requirements. We also use
LoRA to fine-tune the model. The rank is set to 64, and alpha is set to 16.
The training step is 5. We use the AdamW optimizer (Loshchilov & Hutter, 2017)
with a learning rate of 2e-4. The model was fine-tuned using a single A100 GPU
with 4 hours of training time and 7338M of GPU memory.

###  2.4 Baselines and Evaluation Metrics

In our work, we used 14 widely used clinical, biomedical, and general LLMs as
the baseline models, including: six 7B LLMs ( PMC LLaMA 7B Wu et al. (2023),
Medalpaca 7B Han et al. (2023), LLAMA-2 7B Han et al. (2023), Mistral 1*7B
Jiang et al. (2023), Mixtral 8*7B Jiang et al. (2023) , and Bio-Mistral 7B
Labrak et al. (2024) ), one 8B LLM ( LLama3 8B444https://huggingface.co/meta-
llama/Meta-Llama-3-8B), four 13B models (MedLLaMA 13B Wu et al. (2023) , PMC
LLaMA 13B Wu et al. (2023), Medalpaca 13B Han et al. (2023), and LLaMA2 13B
Touvron et al. (2023) ) three 70B LLMs ( LLaMA2 70B Touvron et al. (2023) ,
Llama3-OpenBioLLM-70B555https://huggingface.co/aaditya/Llama3-OpenBioLLM-70B
and ClinicalCamel-70B Toma et al. (2023) ). Same with Li & Huang (2023), we
evaluate all the models based on generative evaluation metrics, including
Exact Match (Rajpurkar et al., 2016) , BLEU-2 (Papineni et al., 2002), and
ROUGE-L (Cohan & Goharian, 2016).

###  2.5 Robustness Testbeds

##### Counterfactual Robustness

Constructing a high-quality annotation corpus is challenging work for
phenotype extraction, as it often involves dealing with incorrect data
labeling. In our work, the mislabeled instances are called counterfactual
instances. In the condition of the mislabeled training dataset, the LLM may
have the ability to avoid negative information. To validate the counterfactual
robustness, same as Li et al. (2024c), we introduced the counterfactual
robustness testbed. Specifically, when constructing the training dataset of
phenotypes extraction, we set the negative rate to be $20\%$, $40\%$, $60\%$
and $80\%$, corresponding to $20\%$, $40\%$, $60\%$ and $80\%$ of instances
being wrongly labeled. An example of incorrect annotation in this dataset
would be entities or entity types that are not present in the input sentence
or are irrelevant to the entity. Subsequently, the input sentence along with
the instruction is fed into the LLM to generate the output.

##### Misspellings Robustness

By checking the clinical notes, we found there are some misspelling words,
such as diagnosis is written to dinosis. To validate the Misspellings
robustness of LLM, we introduced the misspellings robustness testbed.
Specifically, when constructing the training dataset for diagnosis generation
and treatment plan generation, we set the misspelling rate to be $2\%$, $4\%$,
$6\%$, $8\%$, $20\%$, $40\%$, $60\%$ and $80\%$, corresponding to $2\%$,
$4\%$, $6\%$, $8\%$, $20\%$, $40\%$, $60\%$ and $80\%$ of the words in a
sentence being misspelled. Subsequently, the input sentence along with the
instruction is fed into the LLM to generate the output.

##  3 Results

###  3.1 Main Results

| Exact Match | BLEU-2 | ROUGE-L | F1
---|---|---|---|---
Approach | Precision | Recall | F1 | Precision | Recall | F1 | Precision | Recall | F1 | Average
PMC LLaMA 7B | 47.04 | 47.04 | 47.04 | 56.41 | 56.41 | 56.41 | 66.98 | 66.98 | 66.98 | 56.81
Medalpaca 7B | 41.75 | 41.75 | 41.75 | 50.16 | 50.16 | 50.16 | 62.07 | 62.07 | 62.07 | 51.33
LLAMA-2 7B | 33.18 | 33.18 | 33.18 | 42.80 | 42.80 | 42.80 | 55.09 | 55.09 | 55.09 | 43.96
Mistral 1*7B | 45.40 | 45.40 | 45.40 | 54.29 | 54.29 | 54.29 | 65.52 | 65.52 | 65.52 | 55.07
Mixtral 8*7B | 51.32 | 51.32 | 51.32 | 59.35 | 59.35 | 59.35 | 69.70 | 69.70 | 69.70 | 60.12
Bio-Mistral 7B | 62.26 | 62.26 | 62.26 | 68.40 | 68.40 | 68.40 | 76.02 | 76.02 | 76.02 | 68.89
LLama3 8B | 51.60 | 51.60 | 51.60 | 60.13 | 60.13 | 60.13 | 69.97 | 69.97 | 69.97 | 60.57
MedLLaMA 13B | 39.02 | 39.02 | 39.02 | 48.98 | 48.98 | 48.98 | 60.63 | 60.63 | 60.63 | 49.54
PMC LLaMA 13B | 54.97 | 54.97 | 54.97 | 62.56 | 62.56 | 62.56 | 71.06 | 71.06 | 71.06 | 62.86
Medalpaca 13B | 40.66 | 40.66 | 40.66 | 50.58 | 50.58 | 50.58 | 62.40 | 62.40 | 62.40 | 51.21
LLaMA2 13B | 45.76 | 45.76 | 45.76 | 54.70 | 54.70 | 54.70 | 65.80 | 65.80 | 65.80 | 55.42
LLaMA2 70B | 50.23 | 50.23 | 50.23 | 59.63 | 59.63 | 59.63 | 69.25 | 69.25 | 69.25 | 59.70
Llama3-OpenBioLLM-70B | 54.42 | 54.42 | 54.42 | 62.36 | 62.36 | 62.36 | 71.65 | 71.65 | 71.65 | 62.81
ClinicalCamel-70B | 54.60 | 54.60 | 54.60 | 63.34 | 63.34 | 63.34 | 72.73 | 72.73 | 72.73 | 63.55
CancerLLM(Ours) | 83.50 | 83.50 | 83.50 | 86.60 | 86.60 | 86.60 | 90.34 | 90.34 | 90.34 | 86.81

Table 2: Results of Cancer Diagnosis Generation

####  3.1.1 Cancer Diagnosis Generation

To assess the scalability of our model, in this part, we evaluate the
performance of CancerLLM in diagnosis generation task. Table 2 presents the
experiment results of various medical LLMs. We have the following
observations: (1) our CancerLLM significantly outperforms all the strong
baselines and its variants across all evaluation metrics. (2) We observed that
CancerLLM  improve the original Mistral 1*7B, and Bio-Mistral 7B by 28.93%,
and 17.92% respectively, in term of average F1 across Exact Match, BLEU–2 and
ROUGE-L. (3) Our model not only achieves the best performance among 7B models
but also surpasses the performance of medical LLMs with 13B and 70B
parameters. (4) Without domain knowledge, larger parameter LLMs are
ineffective. For instance, Llama3-OpenBioLLM-70B and ClinicalCamel-70B achieve
only 62.81% and 63.55%, separately.

####  3.1.2 Treatment Plan Generation

| Exact Match | BLEU-2 | ROUGE-L | F1
---|---|---|---|---
Approach | Precision | Recall | F1 | Precision | Recall | F1 | Precision | Recall | F1 | Average
PMC LLaMA 7B | 10.61 | 10.61 | 10.61 | 64.52 | 64.52 | 64.52 | 74.08 | 74.08 | 74.08 | 49.74
Medalpaca 7B | 11.11 | 1.11 | 11.11 | 65.26 | 65.26 | 65.26 | 74.69 | 74.69 | 74.69 | 50.35
LLAMA-2 7B | 8.51 | 8.51 | 8.51 | 62.28 | 62.28 | 62.28 | 72.09 | 72.09 | 72.09 | 47.63
Mistral 1*7B | 14.01 | 14.01 | 14.01 | 67.30 | 67.30 | 67.30 | 75.94 | 75.94 | 75.94 | 52.42
Mixtral 8*7B | 12.91 | 12.91 | 12.91 | 65.83 | 65.83 | 65.83 | 74.98 | 74.98 | 74.98 | 51.24
Bio-Mistral 7B | 13.71 | 13.71 | 13.71 | 67.85 | 67.85 | 67.85 | 76.61 | 76.61 | 76.61 | 52.72
LLama3 8B | 12.31 | 12.31 | 12.31 | 65.60 | 65.60 | 65.60 | 74.26 | 74.26 | 74.26 | 50.72
MedLLaMA 13B | 11.91 | 11.91 | 11.91 | 66.48 | 66.48 | 66.48 | 75.59 | 75.59 | 75.59 | 51.33
PMC LLaMA 13B | 12.11 | 12.11 | 12.11 | 66.34 | 66.34 | 66.34 | 75.62 | 75.62 | 75.62 | 51.13
Medalpaca 13B | 12.11 | 12.11 | 12.11 | 65.66 | 65.66 | 65.66 | 74.55 | 74.55 | 74.55 | 50.77
LLaMA2 13B | 9.51 | 9.51 | 9.51 | 64.36 | 64.36 | 64.36 | 73.93 | 73.93 | 73.93 | 49.27
LLaMA2 70B | 14.01 | 14.01 | 14.01 | 66.55 | 66.55 | 66.55 | 75.61 | 75.61 | 75.61 | 52.06
Llama3-OpenBioLLM-70B | 9.01 | 9.01 | 9.01 | 62.76 | 62.76 | 62.76 | 72.54 | 72.54 | 72.54 | 48.10
ClinicalCamel-70B | 14.01 | 14.01 | 14.01 | 67.23 | 67.23 | 67.23 | 75.97 | 75.97 | 75.97 | 52.40
CancerLLM(Ours) | 15.72 | 15.72 | 15.72 | 71.09 | 71.09 | 71.09 | 79.41 | 79.41 | 79.41 | 55.40

Table 3: Results of Cancer Treatment Plan Generation

Table 3 presents the experiment results of various approaches on the treatment
plan generation task. We have the following observations: (1) our CancerLLM
significantly outperforms all the strong baselines and its variants across all
evaluation metrics. (2) We observed that CancerLLM  improve the original
Mistral 1*7B, and Bio-Mistral 7B by 2.59%, and 2.68% respectively, in term of
average F1 across Exact Match, BLEU–2 and ROUGE-L. (4) Without domain
knowledge, larger parameter LLMs still are ineffective. For instance,
Llama3-OpenBioLLM-70B and ClinicalCamel-70B achieve only 48.10% and 52.40%,
separately. (5) An LLM with more parameters is not always better than one with
fewer parameters. For instance, MedLLaMA 13B achieves a performance of 51.33%,
while Llama3-OpenBioLLM-70B only reaches 48.10%. (6) Current medical LLMs
struggle to generate an exact plan that matches the one outlined in clinical
notes created by doctors, as evidenced by the exact match results in Table 3.
Despite our model achieving the best performance, it only attains a 15.72%
Exact Match (F1) score.

| Exact Match | BLEU-2 | ROUGE-L | F1
---|---|---|---|---
Approach | Precision | Recall | F1 | Precision | Recall | F1 | Precision | Recall | F1 | Average
PMC LLaMA 7B | 88.61 | 88.61 | 88.61 | 90.36 | 90.36 | 90.36 | 93.26 | 93.26 | 93.26 | 90.74
Medalpaca 7B | 89.28 | 89.28 | 89.28 | 91.27 | 91.27 | 91.27 | 93.89 | 93.89 | 93.89 | 91.48
LLAMA-2 7B | 89.18 | 89.18 | 89.18 | 90.83 | 90.83 | 90.83 | 93.53 | 93.53 | 93.53 | 91.18
Mistral 1*7B | 89.47 | 89.47 | 89.47 | 91.30 | 91.30 | 91.30 | 94.19 | 94.19 | 94.19 | 91.65
Mixtral 8*7B | 90.23 | 90.23 | 90.23 | 92.05 | 92.05 | 92.05 | 94.50 | 94.50 | 94.50 | 92.26
Bio-Mistral 7B | 88.90 | 88.90 | 88.90 | 91.05 | 91.05 | 91.05 | 93.79 | 93.79 | 93.79 | 91.24
LLama3 8B | 89.94 | 89.94 | 89.94 | 91.75 | 91.75 | 91.75 | 94.34 | 94.34 | 94.34 | 92.01
MedLLaMA 13B | 88.80 | 88.80 | 88.80 | 90.87 | 90.87 | 90.87 | 93.31 | 93.31 | 93.31 | 90.99
PMC LLaMA 13B | 87.95 | 87.95 | 87.95 | 89.64 | 89.64 | 89.64 | 92.58 | 92.58 | 92.58 | 90.06
Medalpaca 13B | 88.61 | 88.61 | 88.61 | 90.37 | 90.37 | 90.37 | 92.94 | 92.94 | 92.94 | 90.64
LLaMA2 13B | 89.85 | 89.85 | 89.85 | 91.54 | 91.54 | 91.54 | 94.21 | 94.21 | 94.21 | 91.86
LLaMA2 70B | 90.04 | 90.04 | 90.04 | 91.62 | 91.62 | 91.62 | 93.98 | 93.98 | 93.98 | 91.88
Llama3-OpenBioLLM-70B | 88.33 | 88.33 | 88.33 | 90.02 | 90.02 | 90.02 | 93.15 | 93.15 | 93.15 | 90.50
ClinicalCamel-70B | 92.02 | 92.02 | 92.02 | 93.62 | 93.62 | 93.62 | 95.52 | 95.52 | 95.52 | 93.72
CancerLLM(Ours) | 89.37 | 89.37 | 89.37 | 91.98 | 91.98 | 91.98 | 93.98 | 93.98 | 93.98 | 91.78

Table 4: Results of Cancer Phenotype Extraction

####  3.1.3 Cancer Phenotype Extraction

Table 4 presents the experiment results of various approaches on the treatment
plan generation task. We have the following observations: (1) Our CancerLLM
significantly outperforms all strong baselines with the same parameter numbers
or number of LLMs across all evaluation metrics. (2) Our CancerLLM performs
comparably with LLMs that have larger parameters. For instance, LLama3 8B
achieves 92.01%, and LLama2 13B achieves 91.86%. (3)CancerLLM could improve
the performance of original Mistral 1*7B, and Bio-Mistral 7B in term of
average F1 across Exact Match, BLEU–2 and ROUGE-L. (4) We observe that
ClinicalCamel-70B achieves the best phenotype extraction performance; however,
it has a larger number of parameters, which affects both training and
inference time, as well as memory consumption. (5) Despite our CancerLLM not
showing significant improvement across all LLMs with different parameters on
the phenotype extraction task, as depicted in Figure 1, our model attains the
best performance across the three tasks examined in this paper.

###  3.2 Results of Robustness Testbeds

####  3.2.1 Counterfactual Robustness

|  | Exact Match | BLEU-2 | ROUGE-L |
---|---|---|---|---|---
Model | rate | Precision | Recall | F1 | Precision | Recall | F1 | Precision | Recall | F1 | Average F1
Mistral 1*7B | 20% | 88.01 | 88.01 | 88.01 | 89.20 | 89.20 | 89.20 | 94.10 | 94.10 | 94.10 | 72.53
40% | 84.82 | 84.82 | 84.82 | 86.85 | 86.85 | 86.85 | 89.58 | 89.58 | 89.58
60% | 77.13 | 77.13 | 77.13 | 79.49 | 79.49 | 79.49 | 82.66 | 82.66 | 82.66
80% | 31.97 | 31.97 | 31.97 | 32.29 | 32.29 | 32.29 | 34.24 | 34.24 | 34.24
CancerLLM(Ours) | 20% | 88.05 | 88.05 | 88.05 | 89.90 | 89.90 | 89.90 | 92.26 | 92.26 | 92.26 |  78.03
40% | 84.54 | 84.54 | 84.54 | 86.71 | 86.71 | 86.71 | 89.56 | 89.56 | 89.56
60% | 77.51 | 77.51 | 77.51 | 79.18 | 79.18 | 79.18 | 82.26 | 82.26 | 82.26
80% | 54.55 | 54.55 | 54.55 | 55.39 | 55.39 | 55.39 | 56.53 | 56.53 | 56.53

Table 5: Counterfactual robustness performance on phenotype extraction. The
rate refers to different counterfactual rate.

To validate the robustness of our model, we propose a counterfactual
robustness testbed designed to simulate incorrect annotations. Table 5
presents the complete results under various counterfactual rates. We observed
that: (1) Our CancerLLM still achieves the best performance when compared to
Mistrial 1*7B. (2) As the rate increases, the model’s performance
deteriorates. (3) When the rate is set as 20%, 40%, and 60%, our model
exhibits similar performance to Mistral 1*7B. However, when the rate is set at
80%, we observe that CancerLLM achieves higher F1 performance, indicating the
robust counterfactual resilience of our model.

####  3.2.2 Misspellings Robustness

|  |  | Exact Match | BLEU-2 | ROUGE-L |
---|---|---|---|---|---|---
| Model | rate | Precision | Recall | F1 | Precision | Recall | F1 | Precision | Recall | F1 | Average F1
Diagnosis Generation | Bio-Mistral 7B | 2% | 0.82 | 0.82 | 0.82 | 10.38 | 10.38 | 10.38 | 25.23 | 25.23 | 25.23 | 10.18
4% | 0.64 | 0.64 | 0.64 | 11.78 | 11.78 | 11.78 | 27.01 | 27.01 | 27.01
6% | 0.00 | 0.00 | 0.00 | 10.23 | 10.23 | 10.23 | 25.25 | 25.25 | 25.25
8% | 0.09 | 0.09 | 0.09 | 9.56 | 9.56 | 9.56 | 23.80 | 23.80 | 23.80
20% | 1.09 | 1.09 | 1.09 | 11.77 | 11.77 | 11.77 | 27.56 | 27.56 | 27.56
40% | 0.18 | 0.18 | 0.18 | 7.40 | 7.40 | 7.40 | 21.48 | 21.48 | 21.48
60% | 0.00 | 0.00 | 0.00 | 3.86 | 3.86 | 3.86 | 19.36 | 19.36 | 19.36
| 80% | 0.00 | 0.00 | 0.00 | 1.14 | 1.14 | 1.14 | 15.87 | 15.87 | 15.87
CancerLLM(Ours) | 2% | 0.09 | 0.09 | 0.09 | 9.88 | 9.88 | 9.88 | 24.67 | 24.67 | 24.67 | 10.66
4% | 0.00 | 0.00 | 0.00 | 10.29 | 10.29 | 10.29 | 25.37 | 25.37 | 25.37
6% | 0.00 | 0.00 | 0.00 | 9.54 | 9.54 | 9.54 | 24.16 | 24.16 | 24.16
8% | 0.27 | 0.27 | 0.27 | 10.61 | 10.61 | 10.61 | 25.81 | 25.81 | 25.81
20% | 0.09 | 0.09 | 0.09 | 11.09 | 11.09 | 11.09 | 27.09 | 27.09 | 27.09
40% | 0.00 | 0.00 | 0.00 | 10.13 | 10.13 | 10.13 | 27.16 | 27.16 | 27.16
60% | 0.00 | 0.00 | 0.00 | 3.75 | 3.75 | 3.75 | 18.42 | 18.42 | 18.42
80% | 0.00 | 0.00 | 0.00 | 1.16 | 1.16 | 1.16 | 16.25 | 16.25 | 16.25
Treatment Plan Generation | Bio-Mistral 7B | 2% | 0.00 | 0.00 | 0.00 | 22.77 | 22.77 | 22.77 | 37.00 | 37.00 | 37.00 | 14.03
4% | 0.00 | 0.00 | 0.00 | 17.64 | 17.64 | 17.64 | 31.61 | 31.61 | 31.61
6% | 0.00 | 0.00 | 0.00 | 18.09 | 18.09 | 18.09 | 32.25 | 32.25 | 32.25
8% | 0.00 | 0.00 | 0.00 | 16.60 | 16.60 | 16.60 | 30.27 | 30.27 | 30.27
20% | 0.00 | 0.00 | 0.00 | 16.62 | 16.62 | 16.62 | 30.29 | 30.29 | 30.29
40% | 0.00 | 0.00 | 0.00 | 14.59 | 14.59 | 14.59 | 27.76 | 27.76 | 27.76
60% | 0.00 | 0.00 | 0.00 | 14.28 | 14.28 | 14.28 | 27.17 | 27.17 | 27.17
CancerLLM(Ours) | 2% | 0.00 | 0.00 | 0.00 | 23.97 | 23.97 | 23.97 | 38.16 | 38.16 | 38.16 | 15.58
4% | 0.00 | 0.00 | 0.00 | 18.00 | 18.00 | 18.00 | 31.92 | 31.92 | 31.92
6% | 0.00 | 0.00 | 0.00 | 19.87 | 19.87 | 19.87 | 34.24 | 34.24 | 34.24
8% | 0.00 | 0.00 | 0.00 | 20.92 | 20.92 | 20.92 | 35.07 | 35.07 | 35.07
20% | 0.00 | 0.00 | 0.00 | 17.34 | 17.34 | 17.34 | 30.47 | 30.47 | 30.47
40% | 0.00 | 0.00 | 0.00 | 18.80 | 18.80 | 18.80 | 32.15 | 32.15 | 32.15
60% | 0.00 | 0.00 | 0.00 | 19.84 | 19.84 | 19.84 | 33.28 | 33.28 | 33.28

Table 6: Misspellings robustness performance on diagnosis generation and
treatment plan generation. The rate refers to different misspelling rates in
each instance.

To validate the robustness of our model, we propose a misspelling robustness
testbed designed to simulate misspelling in the clinical notes. Table 6
presents the complete results and average F1 performance of different
counterfactual rates across exact match, BLEU-2, and ROUGE-L metrics for both
diagnosis generation and treatment generation tasks. Our observations are as
follows: (1) CancerLLM consistently outperforms Bio-Mistral 7B. For the
diagnosis generation task, CancerLLM achieves an average F1 score of 10.66%,
compared to Bio-Mistral 7B’s 10.18%. In the treatment plan generation task,
CancerLLM scores 15.58%, while Bio-Mistral 7B scores 14.03%. For both
diagnosis generation and treatment plan generation tasks, when the
counterfactual rate is set to 2%-80%, the performance is nearly zero.

###  3.3 Results of Generation Time and GPU Memory

| Phenotype Extraction | Treatment Plan Generation | Diagnosis Generation
---|---|---|---
LLMs | F1 | Time | Used Memory | F1 | Time | Used Memory | F1 | Time | Used Memory
Bio-Mistral 7B | 91.24 | 1:06:55 | 5,746 MB | 52.72 | 9:04:56 | 5,786 MB | 68.89 | 1:07:45 | 5,802 MB
Mistral 1*7B | 91.65 | 57:05 | 5,598 MB | 52.42 | 8:57:08 | 5,650MB | 55.07 | 1:07:49 | 5,680 MB
Mixtral 8*7B | 92.26 | 2:01:27 | 25,086 MB | 51.24 | 11:57:42 | 2,5128 MB | 60.12 | 2:16:14 | 25,166 MB
PMC LLaMA 13B | 90.06 | 1:08:59 | 8,208 MB | 51.13 | 12:09:34 | 10,012 MB | 62,86 | 1:19:52 | 9,208 MB
LLaMA2 13B | 91.86 | 1:08:43 | 8,204 MB | 49.27 | 12:08:25 | 10,090 MB | 55.42 | 1:24:17 | 9,254 MB
ClinicalCamel-70B | 93.72 | 2:50:16 | 37,716 MB | 52.40 | 25:58:21 | 37,670 MB | 63.55 | 3:05:37 | 37,67 MB
CancerLLM(Ours) | 91.78 | 1:14:12 | 5,550 MB | 55.40 | 11:40:14 | 5738 MB | 86.81 | 1:26:33 | 5,768 MB

Table 7: Comparation of Generation Time (hours:minutes:seconds) and Used GPU
Memory (Used Memory: Megabyte (MB)) of different LLMs, F1 refers to the
average F1

In Table 7, we compared the generation time on the whole testing set and used
GPU memory on a single A100 across different LLMs. For the Phenotype
Extraction task, we set the maximum input token length to 1500 and the maximum
new token length to 50. For the Treatment Plan Generation task, we set the
maximum input token length to 4000 and the maximum new token length to 500. In
the Diagnosis Generation task, we set the maximum input token length to 1500
and the maximum new token length to 500. We observed that although Mixtral
8*7B, LLaMA2 13B, and ClinicalCamel-70B exhibit better phenotype extraction
performance, they have higher generation time and GPU memory usage. For
example, ClinicalCamel-70B requires 2:50:16 for inference and uses 37,716 MB
of GPU memory, while CancerLLM requires only 1:14:12 for inference and uses
just 5,550 MB of GPU memory.

##  4 Discussion

###  4.1 Cancer Diagnosis Generation

In this task, we primarily pre-train CancerLLM and explore its effectiveness
in the diagnosis generation task by providing relevant information. As shown
in Table 2, the Bio-Mistrial 7B obtains the best performance among all
baseline LLMs. We guess that one reason for this is that Bio-Mistral 7B is
trained on a large pre-processed corpus, such as the PubMed Central corpus
Canese & Weis (2013). Another reason is that the foundational structure of
Mixtral enhances training effectiveness. The Mixtral 8*7B outperforms the
Mixtral 1*7B by more than 5% in average F1 value. However, Mixtral 8*7B is a
Mixture of Experts (MoE) model with 8 experts, making training and inference
time challenging. Fortunately, our CancerLLM achieves the best model
performance through pre-training and fine-tuning with a single expert model.
Not all LLMs with large parameters achieve the best performance. For example,
the 70B models (LLaMA2 70B, Llama3-OpenBioLLM-70B, and ClinicalCamel-70B) do
not outperform the Bio-Mistral 7B, despite being trained on medical corpora.
This indicates that smaller parameter LLMs can achieve better performance. Our
7B model, CancerLLM, which incorporates cancer domain knowledge, also supports
this point. This is particularly important in the medical domain as it reduces
training time and memory usage, providing opportunities for medical
institutions with limited computational resources.

###  4.2 Cancer Treatment Plan Generation

By comparing extract match results of Table 3 and Table 2, we found that LLMs
struggle with treatment plan generation, as even the best model, CancerLLM,
achieves only 15.72% performance. One reason is that the length of the ground
truth treatment plan is longer than the diagnosis, making it difficult to
generate the same treatment plan as created by doctors. Mistral 1*7B is also
the most powerful model among all baselines, outperforming even the 70B models
such as ClinicalCamel-70B. Mistral 1*7B achieves 52.42% average F1
performance, whereas ClinicalCamel-70B achieves only 50.52% average F1. It
also provides evidence that smaller LLMs could achieve better performance
compared to larger models. Increasing model parameters is not the sole method
for enhancing model performance; exploring how to incorporate high-quality
domain knowledge is also crucial.

###  4.3 Cancer Phenotype Extraction

In Table 4, We observed that the ClinicalCamel-70B model outperforms all
baseline models in terms of average F1 value. However, its large parameter
size significantly impacts both training and inference times, making it less
efficient and more resource-intensive. On the other hand, our 7B model,
CancerLLM, shows promising results, performing comparably to both the Mixture
of Experts (MoE) model Mistrial 8*7B and the LLamA2 13B model. This is
particularly noteworthy given CancerLLM’s smaller parameter size, which makes
it more efficient. We suspect that the reason behind CancerLLM’s performance
lies in the specific nature of the clinical notes and pathology reports we
used for training. These documents may not contain sufficient annotation
information, which is crucial for extracting phenotypic data. This highlights
the importance of data quality and annotation in training effective LLM models
for clinical information extraction tasks.

###  4.4 Robustness

The proposed two testbeds were primarily used to evaluate the counterfactual
robustness and misspelling robustness of the LLMs. In the phenotype extraction
task, when the counterfactual rate is set to 20%, we observed that the
performance of CancerLLM and Mixtrual 1*7B does not significantly decrease.
However, when the rate exceeds 60%, there is a notable decline in F1
performance. This indicates that Mixtrual 1*7B and CancerLLM can handle
scenarios with fewer counterfactual instances. However, we found that when
varying the misspelling rate, both the Bio-Mistral 7B and CancerLLM models
experienced a significant decline in performance. This is particularly evident
in the exact match metric, where performance drops close to zero. These
results highlight the critical importance of correct spelling for LLMs to
generate accurate diagnoses and treatment plans, as even minor misspellings
can severely impact the models’ ability to correctly interpret and process
medical information. This underscores the necessity for meticulous data
preprocessing and validation in clinical settings to ensure that the input
data is free of errors, thereby enabling the models to function at their
highest potential and provide reliable outcomes.

###  4.5 Error Analysis

####  4.5.1 Cancer Diagnosis Generation

Error type | Expected output | Error output
---|---|---
Incomplete Generation |  {breast and lung cancer} |  {metastatic breast cancer}
{nsclc t4n2mo-1a} |  {nsclc t4n2mo}
|  {metastatic lung cancer with brain mets} |  {metastatic lung cancer}
Irrelevant Generation |  { dcis (Ductal carcinoma in situ) } |  {invasive ductal carcinoma}
|  {nsclc} |  { lung cancer}
Misspelling |  { metastatic lung cnacer } |  {metastatic lung cancer}
Redundant Generation |  { nsclc} |  {nsclc,t2n0, s/p lobectomy and chemotherapy}
|  {lung cancer} |  { lung cancer with brain mets}
Abbreviation |  { lung ca } |  {lung cancer}

Table 8: Error generation cases of diagnosis generation. Note: Due to privacy
reasons, we did not provide the relevant clinical notes for each error type.

As shown in Table 8, we have comprehensively summarized two primary instances
of error generation observed in CancerLLM. Our observations reveal the
following:

  * •

(1) Misspellings and abbreviations within the clinical notes exert a
significant influence on both the training and evaluation phases of the model.
These linguistic inaccuracies can mislead the model’s learning process and
hinder its ability to accurately generate diagnoses.

  * •

(2) Another notable observation is the model’s tendency to produce incomplete
output in certain scenarios. For instance, when presented with a ground
diagnosis such as metastatic lung cancer with brain metastases, the model may
overlook crucial details such as the presence of brain metastases, This
omission diminishes the comprehensiveness and accuracy of the generated
diagnoses, potentially leading to misinterpretations and suboptimal clinical
decisions.

These findings underscore the importance of meticulous data preprocessing and
model refinement to mitigate the impact of linguistic nuances and enhance the
CancerLLM’s diagnostic capabilities.

####  4.5.2 Cancer Treatment Plan Generation

Error type | Expected output | Error output
---|---|---
Incorrect generation |  {..additional instruction: off of work until November, pt states, she is looking forward to going back } |  {..additional instruction: doing stretches recommended by lymphedema}
Redundant generation |  {1 continue current therapy. 2\. reviewed routine skin cares. 3) eotv on 10/20 } |  {1.continue current therapy. 2\. follow up on 10/27. patient will be leaving one country }

Table 9: Error generation cases of treatment plan generation. For privacy
reasons, we have not provided the relevant clinical notes for each error type.

As shown in Table 9, we mainly summarized two error-generation cases, Our
observations are as follows:

  * •

We observed that CancerLLM sometimes produces identical descriptions before
receiving additional instructions, yet offers different suggestions. For
instance, even though the clinical notes pertain to lymphedema, the doctor’s
recommendation is to take time off work, whereas CancerLLM suggests doing
stretches.

  * •

While doctors provide the treatment plan step by step, CancerLLM sometimes
overlooks crucial steps and generates redundant information. For instance, as
shown in Table 9, CancerLLM ignores step 2, which involves reviewing routine
skin care, and instead focuses on providing additional details about the
patient’s departure from the country.

These findings underscore the challenge of generating meaningful treatment
plans and establishing appropriate evaluation metrics to assess their
effectiveness.

####  4.5.3 Cancer Phenotype Extraction

Error type | Input sentence | Expected output | Error output
---|---|---|---
Redundant Generation |  the cores are infiltrated by ductal carcinoma growing in nests and sheets with areas of residual lumen formation what is the histological type in the given context? |  {ductal carcinoma} |  {ductal carcinoma, residual lumen formation}
Irrelevant Generation |  one requisition slip and labeled right breast 1:00, part b on the container, please describe the tumor location in the given context |  {1:00} |  {it is not a relevant question}
Repeat Generation |  In the text ypt2 ypn1bii mo, stage iib, what is the stage of cancer in the given context? |  {ypn1bii, iib,mo,ypt2} |  {iib,iib,iib,iib,iib, iib,iib,iib}
Incomplete Generation |  In the text er/pr+, her2 negative, please identify the receptors mentioned in the provided context |  {pr, her2, er} |  {pr,er}
In the text sections from the stellate area in the medical breast at the 9-10:00 position show invasive, please describe the tumor location. |  {medical breast at the 9-10:00} |  {9-10:00 position}
Inaccurate Annotation |  In the text ductal carcinoma, solid, intermediate to high nuclear grade 3, what is the grade of cancer? |  {high, intermediate} |  {3}

Table 10: Error generation cases of cancer phenotype extraction

As shown in Table 10, we have identified and summarized six primary error-
generation cases, shedding light on crucial aspects influencing the model’s
performance. Our observations are as follows:

  * •

Redundant Generation: One prevalent issue is the model’s tendency to generate
repetitive and incomplete outputs when confronted with insufficient contextual
information. This limitation hampers the model’s ability to provide
comprehensive and accurate responses.

  * •

Abbreviation Challenges: Abbreviations pose a significant challenge for
CancerLLM, as evidenced by instances such as those outlined in Table 10
(Incomplete generation), where abbreviations like pr and er lead to incomplete
outputs. Resolving this challenge is vital for enhancing the model’s
interpretive accuracy.

  * •

Contextual Misinterpretation: CancerLLM struggles to comprehend questions
accurately within the given context, as illustrated in Table 10 (Redundant
generation), where it misidentifies residual lumen formation as a histological
type. This highlights the need for improved contextual understanding to
prevent such errors.

  * •

Annotation Quality: The quality of annotation data for phenotype extraction
significantly impacts CancerLLM’s learning process. Inaccurate annotations, as
discussed, can impede the model’s ability to learn effectively, underscoring
the importance of providing high-quality annotation data to facilitate robust
learning outcomes.

These findings emphasize the multifaceted nature of the challenges faced by
CancerLLM and underscore the importance of addressing these issues to enhance
its performance and reliability in clinical applications.

##  5 Conclusion

In this study, we introduced CancerLLM, a medical LLM designed specifically
for the cancer domain. We provided three datasets for cancer phenotype
extraction, cancer diagnosis generation, and cancer treatment plan generation.
Additionally, we proposed two testbeds to evaluate the robustness of
CancerLLM. Comparing CancerLLM with 14 widely used LLMs, our results
demonstrate that CancerLLM achieves state-of-the-art performance across three
tasks. Our work provides invaluable insights and tools for further research on
leveraging AI to enhance cancer domain.

##  6 Acknowledgments

This research was supported by the National Institutes of Health’s National
Center for Complementary and Integrative Health grant number R01AT009457,
National Institute on Aging grant number R01AG078154 and National Cancer
Institute grant number R01CA287413. The content is solely the responsibility
of the authors and does not represent the official views of the National
Institutes of Health. We thank support from UMN’s Center for Leanring Health
System Sciences. We thank support from UMN Data Science Initiative seed grant.
This research was supported by the National Institutes of Health’s National
Center for Advancing Translational Sciences, grant UM1TR004405. The content is
solely the responsibility of the authors and does not necessarily represent
the official views of the National Institutes of Health’s National Center for
Advancing Translational Sciences.

##  7 Competing Interests

The authors declare no competing financial or non-financial interests.

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# On function spaces related to H-sober spaces111This research is supported by
the National Natural Science Foundation of China (Nos. 12071199, 11661057) and
the Natural Science Foundation of Jiangxi Province (No. 20192ACBL20045).

Meng Bao<EMAIL_ADDRESS>College of Mathematics, Sichuan University,
Chengdu 610064, China  Xiaoyuan Zhang<EMAIL_ADDRESS>College of
Mathematics, Sichuan University, Chengdu 610064, China
School of Big Data Science, Hebei Finance University, Baoding 071051, China
Xiaoquan Xu<EMAIL_ADDRESS>Fujian Key Laboratory of Granular Computing and
Applications, Minnan Normal University, Zhangzhou 363000, China

###### Abstract

In this paper, we mainly study the function spaces related to H-sober spaces.
For an irreducible subset system H and $T_{0}$ spaces $X$ and $Y$, it is
proved that $Y$ is H-sober iff the function space $\mathbb{C}(X,Y)$ of all
continuous functions $f:X\longrightarrow Y$ equipped with the topology of
pointwise convergence is H-sober iff the function space $\mathbb{C}(X,Y)$
equipped with the Isbell topology is H-sober. One immediate corollary is that
for a $T_{0}$ space $X$, $Y$ is a sober space (resp., $d$-space, well-filtered
space) iff the function space $\mathbb{C}(X,Y)$ equipped with the topology of
pointwise convergence is a sober space (resp., $d$-space, well-filtered space)
iff the function space $\mathbb{C}(X,Y)$ equipped with the the Isbell topology
is a sober space (resp., $d$-space, well-filtered space). It is shown that
$T_{0}$ spaces $X$ and $Y$, if the function space $\mathbb{C}(X,Y)$ equipped
with the compact-open topology is H-sober, then $Y$ is H-sober. The function
space $\mathbb{C}(X,Y)$ equipped with the Scott topology is also discussed.

###### keywords:

Function space; H-sober space; Pointwise convergence topology; Compact-open
topology; Isbell topology; Scott topology

###### MSC:

54C35; 54D99; 06B30, 06B35

††journal:

##  1 Introduction

Function spaces (equipped with certain topologies) are important structures in
topology and domain theory, which was initially introduced by Dana Scott [14].
As a special kind of mathematical structures, domains serve as mathematical
universes within which people can interpret higher-order functional
programming languages and cartesian closed categories of domains (more
generally, certain topological spaces) are appropriate for models of various
typed and untyped lambda-calculi and functional programming languages (see
[3]). Since whether certain properties of topological spaces are preserved
when passing to function spaces is connected with the cartesian closed
category of topological spaces, this question has attracted considerable
attention in domain theory and non-Hausdorff topology, especially for domains
(which are a special kind of topological spaces when endowed with the Scott
topology), sober spaces, $d$-spaces and well-filtered spaces.

There exists a quite satisfactory theory which deals with the cartesian
closedness of domains. Jung [8, 9], Plotkin [12] and Smyth [16] have made
essential contributions to this theory. For topological spaces, it is well-
known that if $X$ is a $T_{0}$ and $Y$ a sober space, then the function space
$\mathbb{C}(X,Y)$ of all continuous functions $f:X\rightarrow Y$ equipped with
the topology of pointwise convergence is sober (see, for example, [3, Exercise
O-5.16]). Furthermore, in [2], it was shown that for any $T_{0}$ space $X$, a
$T_{0}$ space $Y$ is a $d$-space (resp., sober space) iff the the function
space $\mathbb{C}(X,Y)$ equipped with the topology of pointwise convergence is
a $d$-space (resp., sober space). It is known that for a $T_{0}$ space $X$ and
a $d$-space $Y$, the function space $\mathbb{C}(X,Y)$ equipped with the Isbell
topology is a $d$-space (cf. [3, Lemma II-4.3]). Conversely, in [10], the
authors shown that if the function space $\mathbb{C}(X,Y)$ equipped with the
Isbell topology is a $d$-space, then $Y$ is a $d$-space. For the well-
filteredness, it was proved in [10] that for any core compact space $X$ and
well-filtered space $Y$, the function space $\mathbb{C}(X,Y)$ equipped with
the Isbell topology is well-filtered.

In order to provide a uniform approach to $d$-spaces, sober spaces and well-
filtered spaces, Xu [18] introduced the concepts of irreducible subset system
H and H-sober spaces and developed a general framework for dealing with all
these spaces. The irreducible subset systems $\mathcal{D}$, $\mathcal{R}$,
$\mathsf{WD}$ and $\mathsf{RD}$ are four important ones, where for each
$T_{0}$ space $X$, $\mathcal{D}(X)$ is the set of all directed subsets of $X$,
$\mathcal{R}(X)$ is the set of all irreducible subsets of $X$,
$\mathsf{WD}(X)$ is the set of all well-filtered determined subsets of $X$ and
$\mathsf{RD}(X)$ is the set of all Rudin subsets of $X$ (see [18, 19]). So the
$d$-spaces, sober spaces and well-filtered spaces are three special types of
H-sober spaces. It was proved in [18] that for a $T_{0}$ space $X$ and an
H-sober space $Y$, the function space $\mathbb{C}(X,Y)$ equipped with the
topology of pointwise convergence is H-sober. One immediate corollary is that
for a $T_{0}$ space $X$ and a sober space (resp., $d$-space, well-filtered
space) $Y$, the function space $\mathbb{C}(X,Y)$ equipped with the topology of
pointwise convergence is a sober space (resp., $d$-space, well-filtered
space).

In this paper, we mainly study the function spaces related to H-sober spaces.
For an irreducible subset system H and $T_{0}$ spaces $X$ and $Y$, it is
proved that the following three conditions are equivalent:

  1. 1.

$Y$ is H-sober.

  2. 2.

The function space $\mathbb{C}(X,Y)$ equipped with the pointwise convergence
topology is H-sober.

  3. 3.

The function space $\mathbb{C}(X,Y)$ equipped with the Isbell topology is
H-sober.

Applying this result directly to the irreducible subset systems $\mathcal{D}$,
$\mathcal{R}$ and $\mathsf{WD}$ (or $\mathsf{RD}$), we get the conclusion that
for any $T_{0}$ spaces $X$ and $Y$, the following three conditions are
equivalent: (1) $Y$ is a sober space (resp., $d$-space, well-filtered space);
(2) the function space $\mathbb{C}(X,Y)$ equipped with the topology of
pointwise convergence is a sober space (resp., $d$-space, well-filtered
space); (3) the function space $\mathbb{C}(X,Y)$ equipped with the Isbell
topology is a sober space (resp., $d$-space, well-filtered space). This
conclusion improves the result in [10, Corollary 3.12] that for any core
compact space $X$, if $Y$ is well-filtered, then the function space
$\mathbb{C}(X,Y)$ equipped with the Isbell topology is well-filtered. It is
shown that for $T_{0}$ spaces $X$ and $Y$, if the function space
$\mathbb{C}(X,Y)$ equipped with the compact-open topology is H-sober, then $Y$
is H-sober. Therefore, for any $T_{0}$ spaces $X$ and $Y$, $Y$ is a $d$-space
iff the function space $\mathbb{C}(X,Y)$ equipped with the compact-open
topology is a $d$-space. The function space $\mathbb{C}(X,Y)$ equipped with
the Scott topology is also discussed.

##  2 Preliminaries

In this section, we briefly recall some fundamental concepts and notations
that will be used in this paper, more details can be founded in [1, 3, 4].

For a poset $P$ and $A\subseteq P$, let $\mathord{\uparrow}A=\\{x\in P:a\leq
x\mbox{ for some }a\in A\\}$ (dually $\mathord{\downarrow}A=\\{x\in P:x\leq
a\mbox{ for some }a\in A\\})$. For $A=\\{x\\}$, $\mathord{\uparrow}A$ and
$\mathord{\downarrow}A$ are shortly denoted by $\mathord{\uparrow}x$ and
$\mathord{\downarrow}x$ respectively. A subset $A$ is called a _lower set_
(resp., an _upper set_) if $A=\mathord{\downarrow}A$ (resp.,
$A=\mathord{\uparrow}A$). Let $P^{(<\omega)}=\\{F\subseteq P:F\mbox{~{}is a
finite set}\\}$ and $\mathbf{Fin}P=\\{\uparrow F:\emptyset\neq F\in
P^{(<\omega)}\\}$. A subset $D$ of $P$ is _directed_ provided that it is
nonempty and every finite subset of $D$ has an upper bound in $D$. The set of
all directed sets of $P$ is denoted by $\mathcal{D}(P)$. $P$ is said to be a
_directed complete poset_ , a _dcpo_ for short, if every directed subset of
$P$ has the least upper bound in $P$. As in [3], the _upper topology_ on a
poset $P$, generated by the complements of the principal ideals of $P$, is
denoted by $\upsilon(P)$. The upper sets of $P$ form the (_upper_)
_Alexandroff topology_ $\gamma(P)$. The space $\Gamma~{}\\!\\!P=(P,\gamma(P))$
is called the _Alexandroff space_ of $P$. A subset $U$ of a poset $P$ is
called _Scott open_ if $U=\mathord{\uparrow}U$ and $D\cap U\neq\emptyset$ for
all directed sets $D\subseteq P$ with $\vee D\in U$ whenever $\vee D$ exists.
The topology formed by all Scott open sets of $P$ is called the _Scott
topology_ , written as $\sigma(P)$. $\Sigma P=(P,~{}\\!\sigma(P))$ is called
the _Scott space_ of $P$. Clearly, $\Sigma P$ is a $T_{0}$ space.

Given a $T_{0}$ space $X$, we can define a partially order $\leq_{X}$, called
the _specialization order_ , which is defined by $x\leq_{X}y$ iff
$x\in\overline{\\{y\\}}$. Let $\Omega X$ denote the poset $(X,\leq_{X})$.
Clearly, each open set is an upper set and each closed set is a lower set with
respect to the partially order $\leq_{X}$. Unless otherwise stated, throughout
the paper, whenever an order-theoretic concept is mentioned in a $T_{0}$
space, it is to be interpreted with respect to the specialization order. Let
$\mathcal{O}(X)$ (resp., $\mathcal{C}(X)$) be the set of all open subsets
(resp., closed subsets) of $X$ and denote $\mathcal{S}(X)=\\{\\{x\\}:x\in
X\\}$, $\mathcal{D}(X)=\\{D\subseteq X:D$ is a directed set of $X\\}$. A
retract of $X$ is a topological space $Y$ such that there are two continuous
mappings $f:X\rightarrow Y$ and $g:Y\rightarrow X$ such that $f\circ
g=id_{Y}$.

The category of all sets and mappings is denoted by $\mathbf{Set}$ and the
category of all $T_{0}$ spaces with continuous mappings is denoted by
$\mathbf{Top}_{0}$. A $T_{0}$ space $X$ is called a $d$-_space_ (or _monotone
convergence space_) if $X$ (with the specialization order) is a dcpo and
$\mathcal{O}(X)\subseteq\sigma(X)$ (cf.[3]). Clearly, for a dcpo $P$,
$\Sigma~{}\\!\\!P$ is a $d$-space.

One can directly get the following result (cf. [3, 19]).

######  Proposition 2.1.

For a $T_{0}$ space $X$, the following conditions are equivalent:

  1. 1.

$X$ is a $d$-space.

  2. 2.

$\mathcal{D}_{c}(X)=\mathcal{S}_{c}(X)$, that is, for each
$D\in\mathcal{D}(X)$, there exists a _(_ unique _)_ point $x\in X$ such that
$\overline{D}=\overline{\\{x\\}}$.

A nonempty subset $A$ of a $T_{0}$ space $X$ is called _irreducible_ if for
any $\\{F_{1},F_{2}\\}\subseteq\mathcal{C}(X)$, $A\subseteq F_{1}\cup F_{2}$
implies $A\subseteq F_{1}$ or $A\subseteq F_{2}$. We denote by
${\mathsf{Irr}}(X)$ (resp., ${\mathsf{Irr}}_{c}(X)$) the set of all
irreducible (resp., irreducible closed) subsets of $X$. Clearly, every subset
of $X$ that is directed under $\leq_{X}$ is irreducible. A topological space
$X$ is called _sober_ , if for any $F\in{\mathsf{Irr}}_{c}(X)$, there is a
unique point $x\in X$ such that $F=\overline{\\{x\\}}$.

For a topological space $X$, $\mathcal{G}\subseteq 2^{X}$ and $A\subseteq X$,
let $\Diamond_{\mathcal{G}}A=\\{G\in\mathcal{G}:G\cap A\neq\emptyset\\}$ and
$\Box_{\mathcal{G}}A=\\{G\in\mathcal{G}:G\subseteq A\\}$. The sets
$\Diamond_{\mathcal{G}}A$ and $\Box_{\mathcal{G}}A$ will be simply written as
$\Diamond A$ and $\Box A$ respectively if there is no confusion. The _lower
Vietoris topology_ on $\mathcal{G}$ is the topology that has $\\{\Diamond
U:U\in\mathcal{O}(X)\\}$ as a subbase, and the resulting space is denoted by
$P_{H}(\mathcal{G})$. If $\mathcal{G}\subseteq{\mathsf{Irr}}(X)$, then
$\\{\Diamond_{\mathcal{G}}U:U\in\mathcal{O}(X)\\}$ is a topology on
$\mathcal{G}$.

######  Remark 2.2.

Let $X$ be a $T_{0}$ space.

  1. 1.

If $\mathcal{S}_{c}(X)\subseteq\mathcal{G}$, then the specialization order on
$P_{H}(\mathcal{G})$ is the order of set inclusion, and the _canonical
mapping_ $\eta_{X}:X\longrightarrow P_{H}(\mathcal{G})$, given by
$\eta_{X}(x)=\overline{\\{x\\}}$, is an order and topological embedding.

  2. 2.

The space $X^{s}=P_{H}({\mathsf{Irr}}_{c}(X))$ with the canonical mapping
$\eta_{X}:X\longrightarrow X^{s}$ is the _sobrification_ of $X$ (cf. [3, 4]).

For a $T_{0}$ space $X$, let $2^{X}$ be the set of all subsets of $X$. A
subset $A$ of $X$ is called _saturated_ if $A$ equals the intersection of all
open sets containing it (equivalently, $A$ is an upper set in the
specialization order). Let $\mathcal{S}^{u}(X)$ denote the set of all
principal filters, namely, $\mathcal{S}^{u}(X)=\\{\mathord{\uparrow}x:x\in
X\\}$. We denote by $\mathord{\mathsf{K}}(X)$ the poset of nonempty compact
saturated subsets of $X$ with the _Smyth preorder_ , i.e., for
$K_{1},K_{2}\in\mathord{\mathsf{K}}(X)$, $K_{1}\sqsubseteq K_{2}$ iff
$K_{2}\subseteq K_{1}$. A $T_{0}$ space $X$ is called _well-filtered_ , if for
any open set $U$ and any filtered family
$\mathcal{K}\subseteq\mathord{\mathsf{K}}(X)$,
$\bigcap\mathcal{K}{\subseteq}U$ implies $K{\subseteq}U$ for some
$K{\in}\mathcal{K}$. We consider the upper Vietoris topology on
$\mathord{\mathsf{K}}(X)$, generated by the sets $\Box
U=\\{K\in\mathord{\mathsf{K}}(X):K\subseteq U\\}$, where $U$ ranges over the
open subsets of $X$. The resulting space is called the _Smyth power space_ or
_upper space_ of $X$ and denoted by $P_{S}(X)$.

######  Remark 2.3.

Let $X$ be a $T_{0}$ space. Then

  1. 1.

the specialization order on $P_{S}(X)$ is the Smyth order, that is,
$\leq_{P_{S}(X)}=\sqsubseteq$.

  2. 2.

the _canonical mapping_ $\xi_{X}:X\longrightarrow
P_{S}(X),x\mapsto\mathord{\uparrow}x$, is an order and topological embedding
(cf. [13]).

##  3 Topological Rudin Lemma and H-sober spaces

Rudin’s Lemma plays a crucial role in domain theory and is a useful tool in
studying the various aspects of well-filtered spaces and sober spaces. In [5],
Heckman and Keimel presented the following topological variant of Rudin’s
Lemma.

######  Lemma 3.1.

_(Topological Rudin Lemma)_ Let $X$ be a topological space and $\mathcal{A}$
an irreducible subset of the Smyth power space $P_{S}(X)$. Then every closed
set $C\subseteq X$ that meets all members of $\mathcal{A}$ contains a minimal
irreducible closed subset $A$ that still meets all members of $\mathcal{A}$.

For a $T_{0}$ space $X$ and $\mathcal{K}\subseteq\mathord{\mathsf{K}}(X)$, let
$M(\mathcal{K})=\\{A\in\mathcal{C}(X):K\cap A\neq\emptyset\mbox{~{}for
all~{}}K\in\mathcal{K}\\}$ (that is, $\mathcal{K}\subseteq\Diamond A$) and
$m(\mathcal{K})=\\{A\in\mathcal{C}(X):A\mbox{~{}is a minimal menber
of~{}}M(\mathcal{K})\\}$.

######  Definition 3.2.

([15, 19]) Let $X$ be a $T_{0}$ space and $A$ a nonempty subset of $X$.

  1. 1.

$A$ is said to have the _Rudin property_ (which is called _compactly filtered
property_ in [15]), if there exists a filtered family
$\mathcal{K}\subseteq\mathord{\mathsf{K}}(X)$ such that $\overline{A}\in
m(\mathcal{K})$ (that is, $\overline{A}$ is a minimal closed set that
intersects all members of $\mathcal{K}$). Let $\mathsf{RD}(X)=\\{A\subseteq
X:A\mbox{{~{}has Rudin property}}\\}$ and
$\mathsf{RD}_{c}(X)=\\{\overline{A}:A\in\mathsf{RD}(X)\\}$. The sets in
$\mathsf{RD}(X)$ will also be called _Rudin sets_. Obviously, a subset $A$ of
a space $X$ has Rudin property iff $\overline{A}$ has Rudin property.

  2. 2.

$A$ is called a _well-filtered determined set_ if for any continuous mapping
$f:X\longrightarrow Y$ into a well-filtered space $Y$, there exists a unique
$y_{A}\in Y$ such that $\overline{f(A)}=\overline{\\{y_{A}\\}}$. Denote by
$\mathsf{WD}(X)$ the set of all well-filtered determined subsets of $X$ and
$\mathsf{WD}_{c}(X)=\\{\overline{A}:A\in\mathsf{WD}(X)\\}$. Obviously, a
subset $A$ of a space $X$ is well-filtered determined iff $\overline{A}$ is
well-filtered determined.

######  Proposition 3.3.

_([19, Proposition 6.2])_ Let $X$ be a $T_{0}$ space. Then
$\mathcal{S}(X)\subseteq\mathcal{D}(X)\subseteq\mathsf{RD}(X)\subseteq\mathsf{WD}(X)\subseteq{\mathsf{Irr}}(X)$.

######  Proposition 3.4.

_([15, Lemma 2.5], [19, Lemma 6.23])_ Let $X,Y$ be two $T_{0}$ spaces and
$f:X\longrightarrow Y$ a continuous mapping. Then the following statements
hold:

  1. 1.

If $A\in\mathsf{WD}(X)$, then $f(A)\in\mathsf{WD}(Y)$.

  2. 2.

If $A\in\mathsf{RD}(X)$, then $f(A)\in\mathsf{RD}(Y)$.

Based on the Rudin sets and well-filtered determined sets, the following
characterization of well-filtered spaces was given in [19].

######  Proposition 3.5.

_([19, Corollary 7.11])_ Let $X$ be a $T_{0}$ space. Then the following
conditions are equivalent:

  1. 1.

$X$ is well-filtered.

  2. 2.

$\mathsf{WD}_{c}(X)=\mathcal{S}_{c}(X)$, that is, for any
$A\in\mathsf{WD}(X)$, there exists a unique $a\in X$ such that
$\overline{A}=\overline{\\{a\\}}$.

  3. 3.

$\mathsf{RD}_{c}(X)=\mathcal{S}_{c}(X)$, that is, for any
$A\in\mathsf{RD}(X)$, there exists a unique $a\in X$ such that
$\overline{A}=\overline{\\{a\\}}$.

In order to provide a uniform approach to $d$-spaces, sober spaces and well-
filtered spaces and develop a general framework for dealing with all these
spaces, Xu [18] introduced the following concepts.

######  Definition 3.6.

([18]) (1) A covariant functor ${\rm
H}:\mathbf{Top}_{0}\longrightarrow\mathbf{Set}$ is called a _subset system_ on
$\mathbf{Top}_{0}$ provided that the following two conditions are satisfied:

  1. 1.

$\mathcal{S}(X)\subseteq{\rm H}(X)\subseteq 2^{X}$ (the set of all subsets of
$X$) for each $X\in$ _ob_($\mathbf{Top}_{0}$).

  2. 2.

For any continuous mapping $f:X\longrightarrow Y$ in $\mathbf{Top}_{0}$, ${\rm
H}(f)(A)=f(A)\in{\rm H}(Y)$ for all $A\in{\rm H}(X)$.

(2) A subset system ${\rm H}:\mathbf{Top}_{0}\longrightarrow\mathbf{Set}$ is
called an _irreducible subset system_ , or an _R-subset system_ for short, if
${\rm H}(X)\subseteq{\mathsf{Irr}}(X)$ for all $X\in$
_ob_($\mathbf{Top}_{0}$).

In what follows, the capital letter H always stands for an R-subset system
${\rm H}:\mathbf{Top}_{0}\longrightarrow\mathbf{Set}$. For a $T_{0}$ space
$X$, let ${\rm H}_{c}(X)=\\{\overline{A}:A\in{\rm H}(X)\\}$. Define a
partially order $\leq$ on the set of all R-subset systems by ${\rm
H}_{1}\leq{\rm H}_{2}$ iff ${\rm H}_{1}(X)\subseteq{\rm H}_{2}(X)$ for all
$T_{0}$ spaces $X$.

Here are some important examples of R-subset systems used later:

  1. 1.

$\mathcal{S}$ (for $X\in$ _ob_($\mathbf{Top}_{0}$), $\mathcal{S}(X)$ is the
set of all single point subsets of $X$).

  2. 2.

$\mathcal{D}$ (for $X\in$ _ob_($\mathbf{Top}_{0}$), $\mathcal{D}(X)$ is the
set of all directed subsets of $X$).

  3. 3.

$\mathcal{R}$ (for $X\in$ _ob_($\mathbf{Top}_{0}$), $\mathcal{R}(X)$ is the
set of all irreducible subsets of $X$).

  4. 4.

$\mathsf{WD}$ (for $X\in$ _ob_($\mathbf{Top}_{0}$), $\mathsf{WD}(X)$ is the
set of all well-filtered determined subsets of $X$).

  5. 5.

$\mathsf{RD}$ (for $X\in$ _ob_($\mathbf{Top}_{0}$), $\mathsf{RD}(X)$ is the
set of all Rudin subsets of $X$).

By Propositions 3.3 and 3.4, $\mathsf{WD}$ and $\mathsf{RD}$ are R-subset
systems; and by Proposition 3.3, we have
$\mathcal{S}\leq\mathcal{D}\leq\mathsf{RD}\leq\mathsf{WD}\leq\mathcal{R}$.

######  Definition 3.7.

([18]) Let ${\rm H}:\mathbf{Top}_{0}\longrightarrow\mathbf{Set}$ be an
R-subset system and $X$ be a $T_{0}$ space. $X$ is called _H_ -_sober_ if for
any $A\in{\rm H}(X)$, there is a (unique) point $x\in X$ such that
$\overline{A}=\overline{\\{x\\}}$ or, equivalently, if ${\rm
H}_{c}(X)=\mathcal{S}_{c}(X)$.

######  Definition 3.8.

([18]) Let ${\rm H}:\mathbf{Top}_{0}\longrightarrow\mathbf{Set}$ be an
R-subset system and $X$ a $T_{0}$ space. A subset $A$ of $X$ is called _H_
-_sober determined_ , if for any continuous mapping $f:X\longrightarrow Y$ to
an H-sober space $Y$, there exists a unique $y_{A}\in Y$ such that
$\overline{f(A)}=\overline{\\{y_{A}\\}}$. Denote by ${\rm H}^{d}(X)$ the set
of all H-sober determined subsets of $X$. The set of all closed H-sober
determined subsets of $X$ is denoted by ${\rm H}^{d}_{c}(X)$.

##  4 Function spaces related to H-sober spaces

For $T_{0}$ spaces $X$ and $Y$, there are four important topologies on the set
$\mathbb{C}(X,Y)$ of all continuous functions from $X$ to $Y$, namely, the
pointwise convergence topology, compact-open topology, Isbell topology and
Scott topology. In this section, we will discuss all of these topologies on
$\mathbb{C}(X,Y)$.

We begin by recalling the definitions of pointwise convergence topology,
Isbell topology and compact-open topology. For further details, we refer the
reader to [1, 3, 4].

In the following, $\mathbb{C}(X,Y)$ always means the set of all continuous
functions from $X$ to $Y$.

######  Definition 4.1.

Let $X$ and $Y$ be topological spaces.

  1. 1.

For a point $x\in X$ and an open set $U\in\mathcal{O}(Y)$, let
$S(x,U)=\\{f\in\mathbb{C}(X,Y)$ : $f(x)\in U\\}$. The set $\\{S(x,U):x\in
X,U\in\mathcal{O}(Y)\\}$ is a subbasis for the _pointwise convergence
topology_ (i.e., the relative product topology) on $\mathbb{C}(X,Y)$. Let
$[X\rightarrow Y]_{P}$ denote the function space $\mathbb{C}(X,Y)$ endowed
with the topology of pointwise convergence.

  2. 2.

The _Isbell topology_ on the set $\mathbb{C}(X,Y)$ is generated by the subsets
of the form $N(\mathcal{H}\leftarrow
V)=\\{f\in\mathbb{C}(X,Y):f^{-1}(V)\in\mathcal{H}\\},$ where $\mathcal{H}$ is
a Scott open subset of the complete lattice $\mathcal{O}(X)$ and $V$ is open
in $Y$. Let $[X\rightarrow Y]_{I}$ denote the function space $\mathbb{C}(X,Y)$
endowed with the Isbell topology.

  3. 3.

The _compact-open topology_ on the set $\mathbb{C}(X,Y)$ is generated by the
subsets of the form $N(K\rightarrow V)=\\{f\in\mathbb{C}(X,Y):f(K)\subseteq
V\\},$ where $K$ is compact in $X$ and $V$ is open in $Y$. Let $[X\rightarrow
Y]_{K}$ denote the function space $\mathbb{C}(X,Y)$ endowed with the compact-
open topology.

For a topological space $X$ and a $T_{0}$ space $Y$, $Y$ (with the
specialization order) is a poset, whence $\mathbb{C}(X,Y)$ is a poset with the
pointwise order. Denote by $[X\rightarrow Y]_{\Sigma}$ the Scott space
$\Sigma\mathbb{C}(X,Y)$. It is easy to see that the specialization orders on
$[X\rightarrow Y]_{P}$, $[X\rightarrow Y]_{K}$, $[X\rightarrow Y]_{I}$ and
$[X\rightarrow Y]_{\Sigma}$ are all the usually pointwise order on
$\mathbb{C}(X,Y)$, i.e., for any $f,g\in\mathbb{C}(X,Y)$,

$f\leq_{[X\rightarrow Y]_{P}}g~{}\mbox{iff}~{}f\leq_{[X\rightarrow
Y]_{K}}g~{}\mbox{iff}~{}f\leq_{[X\rightarrow
Y]_{I}}g~{}\mbox{iff}~{}f\leq_{[X\rightarrow
Y]_{\Sigma}}g~{}\mbox{iff}~{}f(x)\leq_{Y}g(x)~{}\mbox{for all}~{}x\in X.$

######  Lemma 4.2.

_([3, Lemma II-4.2])_ For $T_{0}$ spaces $X$ and $Y$, the Isbell topology is
finer than the compact-open topology which in turn is finer than the topology
of pointwise convergence. If $X$ is sober and $\mathcal{O}(X)$ is a continuous
lattice, then the Isbell topology and the compact-open topology agree.

######  Lemma 4.3.

Let $X,Y$ be $T_{0}$ spaces and $x\in X$. Consider the function

$\varphi^{P}_{x}:[X\rightarrow Y]_{P}\longrightarrow Y,$
$\\!~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}f\\!~{}~{}~{}\mapsto\\!~{}f(x).$

Then $\varphi^{P}_{x}$ is continuous.

###### Proof.

Suppose $f\in[X\rightarrow Y]_{P}$ and $\varphi^{P}_{x}(f)\in V$ (i.e.,
$f(x)\in V$), where $V$ is open in $Y$. Then $f\in S(x,V)$ and $S(x,V)$ is
open in $[X\rightarrow Y]_{P}$. Clearly, $\varphi^{P}_{x}(S(x,V))\subseteq V$.
So $\varphi^{P}_{x}:[X\rightarrow Y]_{P}\longrightarrow Y$ is continuous. ∎

######  Corollary 4.4.

Let $X,Y$ be $T_{0}$ spaces and $x\in X$. Then

  1. 1.

the function $\varphi^{K}_{x}:[X\rightarrow Y]_{K}\longrightarrow Y$,
$f\mapsto f(x)$, is continuous.

  2. 2.

the function $\varphi^{I}_{x}:[X\rightarrow Y]_{I}\longrightarrow Y$,
$f\mapsto f(x)$, is continuous.

###### Proof.

By Lemmas 4.2 and 4.3. ∎

In the following, for topological spaces $X,Y$ and $y\in Y$, $c_{y}$ denotes
the constant function from $X$ to $Y$ with value $y$, i.e., $c_{y}(x)=y$ for
all $x\in X$.

######  Lemma 4.5.

_([10, Lemma 3.2])_ Let $X$ and $Y$ be $T_{0}$ spaces. Consider the function

$\\!~{}~{}~{}~{}~{}\psi^{I}:Y\longrightarrow[X\rightarrow Y]_{I},$
$y\\!~{}\mapsto\\!~{}c_{y}.$

Then $\psi^{I}$ is continuous.

By Lemmas 4.2 and 4.5, we can directly obtain the following corollary.

######  Corollary 4.6.

Let $X$ and $Y$ be $T_{0}$ spaces. Then

  1. 1.

the function $\psi^{K}:Y\longrightarrow[X\rightarrow Y]_{K}$, $y\mapsto
c_{y}$, is continuous.

  2. 2.

the function $\psi^{P}:Y\longrightarrow[X\rightarrow Y]_{P}$, $y\mapsto
c_{y}$, is continuous.

######  Remark 4.7.

The similar result of Lemma 4.5 and Corollary 4.6 for the Scott topology does
not hold in general. Indeed, let $X=1=\\{0\\}$ be the topological space with
only one point (clearly, $\mathcal{O}(X)=\mathcal{O}(1)=\\{\emptyset,1\\}$ and
$Y$ the space $(P,\upsilon(P))$, where $P$ is a poset for which the Scott
topology is strictly finer than the upper topology on $P$ (for example, $P$ is
a countably infinite set with the discrete order). Then
$\mathbb{C}(X,Y)=\\{c_{y}:y\in Y\\}$ and $c_{y}\mapsto y:[X\rightarrow
Y]_{\Sigma}\rightarrow\Sigma Y=\Sigma P$ is a homeomorphism. Since
$\upsilon(P)\subsetneqq\sigma(P)$, the function
$\psi^{\Sigma}:Y\longrightarrow[X\rightarrow Y]_{\Sigma}$, $y\mapsto c_{y}$,
is not continuous.

######  Lemma 4.8.

_([18, Proposition 4.28])_ A retract of an H-sober space is H-sober.

######  Corollary 4.9.

A retract of a sober space (resp., $d$-space, well-filtered space) is a sober
space (resp., $d$-space, well-filtered space).

######  Proposition 4.10.

Let $X,Y$ be $T_{0}$ spaces.

  1. 1.

If the function space $\mathbb{C}(X,Y)$ equipped with the compact-open
topology is H-sober, then $Y$ is H-sober.

  2. 2.

If the function space $\mathbb{C}(X,Y)$ equipped with the pointwise
convergence topology is H-sober, then $Y$ is H-sober.

###### Proof.

(1): Suppose that $[X\rightarrow Y]_{K}$ is H-sober. Select an $x\in X$. Then
for any $y\in Y$, $(\varphi^{K}_{x}\circ\psi^{K})(y)=\varphi^{K}_{x}(c_{y})=y$
and $\varphi^{K}_{x}$, $\psi^{K}$ are continuous by Corollary 4.4 (1) and
Corollary 4.6 (1). Thus $Y$ is a retract of $[X\rightarrow Y]_{K}$. Hence $Y$
is H-sober by Lemma 4.8.

(2): The proof is similar to that of (1).

∎

######  Theorem 4.11.

_([18, Theorem 4.30])_ Let $H:\mathbf{Top}_{0}\longrightarrow\mathbf{Set}$ be
an R-subset system, $X$ a $T_{0}$ space and $Y$ an H-sober space. Then the
function space $\mathbb{C}(X,Y)$ equipped with the topology of pointwise
convergence is H-sober.

By Proposition 4.10 and Theorem 4.11, we have the following result.

######  Corollary 4.12.

Let $H:\mathbf{Top}_{0}\longrightarrow\mathbf{Set}$ be an R-subset system and
$X,Y$ two $T_{0}$ spaces. Then the following two conditions are equivalent:

  1. 1.

$Y$ is H-sober.

  2. 2.

The function space $\mathbb{C}(X,Y)$ equipped with the topology of pointwise
convergence is H-sober.

Applying Proposition 4.10 and Corollary 4.12 directly to the R-subset systems
$\mathcal{D}$, $\mathcal{R}$ and $\mathsf{WD}$ (or $\mathsf{RD}$), we get the
following two corollaries.

######  Corollary 4.13.

Let $X,Y$ be $T_{0}$ spaces. If the function space $\mathbb{C}(X,Y)$ equipped
with the compact-open topology is a sober space (resp., $d$-spaces, well-
filtered space), then $Y$ is a sober space (resp., $d$-spaces, well-filtered
space).

######  Corollary 4.14.

For $T_{0}$ spaces $X$ and $Y$, the following two conditions are equivalent:

  1. 1.

$Y$ is a sober space (resp., $d$-spaces, well-filtered space).

  2. 2.

The function space $\mathbb{C}(X,Y)$ equipped with the pointwise convergence
topology is a sober space (resp., $d$-space, well-filtered space).

The results for $d$-spaces and sober spaces in Corollary 4.14 were first shown
in [2, Theorem 3 and Theorem 6] by a different method.

######  Proposition 4.15.

([3, LemmaII-4.3]) For a $T_{0}$ $X$ and a $d$-space $Y$, the function space
$\mathbb{C}(X,Y)$ equipped with the Isbell topology is a $d$-space and hence
the Scott topology on $\mathbb{C}(X,Y)$ is finer than the Isbell topology.

From Lemma 4.2 and Proposition 4.15 we deduce the following result.

######  Corollary 4.16.

For a $T_{0}$ space $X$ and a $d$-space $Y$, the function space
$\mathbb{C}(X,Y)$ equipped with the compact-open topology is a $d$-space.

By Corollary 4.13 and Corollary 4.16, we get the following.

######  Corollary 4.17.

For $T_{0}$ spaces $X$ and $Y$, the following two conditions are equivalent:

  1. 1.

$Y$ is a $d$-space.

  2. 2.

The function space $\mathbb{C}(X,Y)$ equipped with the compact-open topology
is a $d$-space.

######  Remark 4.18.

We can give a direct proof of Corollary 4.16.

###### Proof.

Suppose that $X$ is a $T_{0}$ space and $Y$ is a $d$-space. We show that the
function space $\mathbb{C}(X,Y)$ equipped with the compact-open topology is a
$d$-space.

Let $\mathcal{F}\in\mathcal{D}([X\rightarrow Y]_{K})$. Since the
specialization order on $[X\rightarrow Y]_{K}$ is the usually pointwise order
on $\mathbb{C}(X,Y)$, we have that for each $x\in X$,
$\\{f(x):f\in\mathcal{F}\\}\in\mathcal{D}(Y)$. As $Y$ is a $d$-space, there
exists a unique point $a_{x}\in Y$ such that
$\overline{\\{f(x):f\in\mathcal{F}\\}}=\overline{\\{a_{x}\\}}$. Then we can
define a function $g:X\longrightarrow Y$ by $g(x)=a_{x}$ for each $x\in X$. It
is straightforward to verify that $g$ is continuous (see Claim 1 in the proof
of Theorem 4.24 below). Whence $g$ is an upper bound of $\mathcal{F}$ in
$[X\rightarrow Y]_{K}$, and consequently, ${\rm cl}_{[X\rightarrow
Y]_{I}}\mathcal{F}\subseteq{\rm cl}_{[X\rightarrow Y]_{I}}\\{g\\}$.

Now we show that $g\in{\rm cl}_{[X\rightarrow Y]_{K}}\mathcal{F}$. Let
$N(K\rightarrow V)$ be a subbasis open set in $[X\rightarrow Y]_{K}$ with
$g\in N(K\rightarrow V)$, where $K$ is a compact subset of $X$ and $V$ is an
open subset of $Y$. Then for any $x\in K$, $g(x)\in V$. As
$\overline{\\{f(x):f\in\mathcal{F}\\}}=\overline{\\{g(x)\\}}$, there exists
$f\in\mathcal{F}$ with $f(x)\in V$, whence $x\in f^{-1}(V)$. Thus
$K\subseteq\bigcup_{f\in\mathcal{F}}f^{-1}(V)$. By the compactness of $K$,
there exists $\\{f_{1},f_{2},\cdots,f_{n}\\}\subseteq F$ such that
$K\subseteq\bigcup_{i=1}^{n}f_{i}^{-1}(V)$. Since $\mathcal{F}$ is directed in
$[X\rightarrow Y]_{K}$, there is $h\in\mathcal{F}$ such that $f_{i}\leq h$ for
all $i\in\\{1,2,\cdots,n\\}$. Therefore,
$K\subseteq\bigcup_{i=1}^{n}f_{i}^{-1}(V)\subseteq h^{-1}(V)$ (note that open
sets in $Y$ are upper sets with the specialization order of $Y$), i.e., $h\in
N(K\rightarrow V)$. So $\mathcal{F}\cap N(K\rightarrow V)\neq\emptyset$. Since
$\mathcal{F}\in\mathcal{D}([X\rightarrow
Y]_{K})\subseteq{\mathsf{Irr}}([X\rightarrow Y]_{K})$ and all basic open sets
of $g$ in $[X\rightarrow Y]_{K}$ must meet $\mathcal{F}$, we get that
$g\in{\rm cl}_{[X\rightarrow Y]_{K}}\mathcal{F}$, and hence ${\rm
cl}_{[X\rightarrow Y]_{K}}\mathcal{F}={\rm cl}_{[X\rightarrow Y]_{K}}\\{g\\}$.
Therefore, the function space $\mathbb{C}(X,Y)$ equipped with the compact-open
topology is a $d$-space.

∎

But we do not know whether the converse of Proposition 4.10 (1) holds
(especially, whether Corollary 4.16 holds for well-filtered spaces and sober
spaces). Therefore, we pose the following questions.

######  Question 4.19.

For a $T_{0}$ space $X$ and an H-sober space $Y$, is the function space
$\mathbb{C}(X,Y)$ equipped with the compact-open topology H-sober?

######  Question 4.20.

For a $T_{0}$ space $X$ and a sober (resp., well-filtered) space $Y$, is the
function space $\mathbb{C}(X,Y)$ equipped with the compact-open topology sober
(resp., well-filtered)?

######  Proposition 4.21.

For $T_{0}$ spaces $X$ and $Y$, if the function space $\mathbb{C}(X,Y)$
equipped with the Isbell topology is H-sober, then $Y$ is H-sober.

###### Proof.

Suppose that $[X\rightarrow Y]_{\Sigma}$ is H-sober. Select an $x\in X$. Then
for any $y\in Y$, $(\varphi^{I}_{x}\circ\psi^{I})(y)=\varphi^{I}_{x}(c_{y})=y$
and $\varphi^{I}_{x}$, $\psi^{I}$ are continuous by Corollary 4.4 (2) and
Lemma 4.5. Thus $Y$ is a retract of $[X\rightarrow Y]_{I}$. Hence $Y$ is
H-sober by Lemma 4.8. ∎

######  Lemma 4.22.

Let $X$, $Y$ be topological spaces and $V$ is open in $Y$. Consider the
function

$\zeta_{V}:[X\rightarrow Y]_{I}\longrightarrow\Sigma\mathcal{O}(X),$
$\\!~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}f\\!~{}~{}~{}\mapsto\\!~{}f^{-1}(V).$

Then $\zeta_{V}$ is continuous.

###### Proof.

Suppose $V\in\mathcal{O}(Y)$, $f\in[X\rightarrow Y]_{I}$ and
$\zeta_{V}(f)=f^{-1}(V)\in\mathcal{H}$, where $\mathcal{H}$ is Scott open in
$\mathcal{O}(X)$. Then $f\in N(\mathcal{H}\leftarrow V)$,
$N(\mathcal{H}\leftarrow V)$ is open in $[X\rightarrow Y]_{I}$ and for any
$g\in N(\mathcal{H}\leftarrow V)$, $\zeta_{V}(g)=g^{-1}(V)\in\mathcal{H}$,
whence $\zeta_{V}(N(\mathcal{H}\leftarrow V))\subseteq\mathcal{H}$. It follows
that $\zeta_{V}:[X\rightarrow Y]_{I}\longrightarrow\Sigma\mathcal{O}(X)$ is
continuous. ∎

######  Lemma 4.23.

([18]) Let $H:\mathbf{Top}_{0}\longrightarrow\mathbf{Set}$ be an R-subset
system. Then for any $T_{0}$ space $X$, $P_{H}(H_{c}^{d}(X))$ is an H-sober
space. In fact, for any $\\{A_{i}:i\in I\\}\in H(P_{H}(H_{c}^{d}(X)))$,
$A=\overline{\bigcup_{i\in I}A_{i}}\in H_{c}^{d}(X)$ and
$\overline{\\{A_{i}:i\in I\\}}=\overline{\\{A\\}}$ in $P_{H}(H_{c}^{d}(X))$.
Moreover, $X^{h}=P_{H}(H_{c}^{d}(X))$ with the canonical topological embedding
$\eta_{X}^{h}:X\longrightarrow X^{h}$ is the H-sobrification of $X$, where
$\eta_{X}^{h}(x)=\overline{\\{x\\}}$ for all $x\in X$.

One of the main results of this paper is the following.

######  Theorem 4.24.

Let $H:\mathbf{Top}_{0}\longrightarrow\mathbf{Set}$ be an R-subset system.
Then for a $T_{0}$ space $X$ and an H-sober space $Y$, the function space
$\mathbb{C}(X,Y)$ equipped with the Isbell topology is an H-sober space.

###### Proof.

Let $\mathcal{F}\in H([X\rightarrow Y]_{I})$. Since the specialization order
on $[X\rightarrow Y]_{I}$ is the usually pointwise order on $\mathbb{C}(X,Y)$,
$[X\rightarrow Y]_{I}$ is $T_{0}$. For $x\in X$, by Corollary 4.4 (2), the
function $\varphi^{I}_{x}:[X\rightarrow Y]_{I}\longrightarrow Y$, $f\mapsto
f(x)$, is continuous. So
$H(\varphi^{I}_{x})(\mathcal{F})=\varphi^{I}_{x}(\mathcal{F})=\\{f(x):f\in\mathcal{F}\\}\in
H(Y)$. As $Y$ is H-sober, there is a unique point $a_{x}\in Y$ such that
$\overline{\\{f(x):f\in\mathcal{\mathcal{F}}\\}}=\overline{\\{a_{x}\\}}$. Now
we can define a function

$g:X\longrightarrow Y\ \ by\ \ g(x)=a_{x}\hbox{~{}for~{}each~{}}x\in X.$

Claim 1: $g:X\longrightarrow Y$ is continuous.

Let $x\in X$ and $V\in\mathcal{O}(Y)$ with $g(x)=a_{x}\in V$. Then by
$\overline{\\{a_{x}\\}}=\overline{\\{f(x):f\in\mathcal{\mathcal{F}}\\}}=$, we
have $\\{f(x):f\in\mathcal{F}\\}\bigcap V\neq\emptyset$, whence there is
$f\in\mathcal{F}$ such that $f(x)\in V$. As $f:X\longrightarrow Y$ is
continuous, there is a $U\in\mathcal{O}(X)$ with $x\in U$ and $f(U)\subseteq
V$. For any $z\in U$,
$f(z)\in\overline{\\{f(z):f\in\mathcal{F}\\}}=\overline{\\{a_{z}\\}}=\overline{\\{g(z)\\}}$,
and hence $g(z)\in V$ (note that $V$ is an upper set in $Y$ with the
specialization order). So $g(U)\subseteq V$. Thus $g$ is continuous.

Claim 2: For any subbasis open set $N(\mathcal{H}\leftarrow W)$ in
$[X\rightarrow Y]_{I}$ with $g\in N(\mathcal{H}\leftarrow W)$, where
$\mathcal{H}$ is a Scott open subset of $\mathcal{O}(X)$ and $W$ is an open
subset of $Y$, we have $\mathcal{F}\bigcap N(\mathcal{H}\leftarrow
W)\neq\emptyset$.

Since $g\in N(\mathcal{H}\leftarrow W)$, $W\in\mathcal{O}(Y))$, we have
$g^{-1}(W)\in\mathcal{H}$. For any $x\in g^{-1}(W)$ (i.e., $g(x)\in W$), by
$g(x)=a_{x}\in\overline{\\{f(x):f\in\mathcal{F}\\}}$, there exists
$f_{0}\in\mathcal{F}$ such that $f_{0}(x)\in W$. Then $x\in
f_{0}^{-1}(W)\subseteq\bigcup_{f\in\mathcal{F}}f^{-1}(W)$. So
$g^{-1}(W)\subseteq\bigcup_{f\in\mathcal{F}}f^{-1}(W)$. As $\mathcal{H}$ is an
upper set, we have $\bigcup_{f\in\mathcal{F}}f^{-1}(W)\in\mathcal{H}$.

Since $\Sigma\mathcal{O}(X)$ is $T_{0}$, one can directly deduce that
$P_{H}(H_{c}^{d}(\Sigma\mathcal{O}(X)))$ is $T_{0}$. By Lemmas 4.22 and 4.23,
for each continuous function $f:X\longrightarrow Y$,

$\zeta_{W}:[X\rightarrow Y]_{I}\longrightarrow\Sigma\mathcal{O}(X),$
$\\!~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}f\\!~{}~{}~{}\mapsto\\!~{}f^{-1}(W),$

is continuous, and for any $U\in\mathcal{O}(X)$,

$\eta_{\Sigma\mathcal{O}(X)}^{h}:\Sigma\mathcal{O}(X)\longrightarrow
P_{H}(H_{c}^{d}(\Sigma\mathcal{O}(X))),$
$\\!~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}U\\!~{}~{}\mapsto\\!~{}~{}{\rm
cl}_{\Sigma\mathcal{O}(X)}\\{U\\},$

is continuous. So for each continuous function $f:X\longrightarrow Y$,

$\eta_{\Sigma\mathcal{O}(X)}^{h}\circ\zeta_{W}:[X\rightarrow
Y]_{I}\longrightarrow P_{H}(H_{c}^{d}(\Sigma\mathcal{O}(X))),$
$\\!~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}f\\!~{}~{}~{}\mapsto\\!~{}{\rm
cl}_{\Sigma\mathcal{O}(X)}\\{f^{-1}(W)\\},$

is continuous. Thus $\\{{\rm
cl}_{\Sigma\mathcal{O}(X)}\\{f^{-1}(W)\\}:f\in\mathcal{F}\\}\in
H(P_{H}(H_{c}^{d}(\Sigma\mathcal{O}(X))))$. By Lemma 4.23 again,

${\rm cl}_{\Sigma\mathcal{O}(X)}\bigcup_{f\in\mathcal{F}}{\rm
cl}_{\Sigma\mathcal{O}(X)}\\{f^{-1}(W)\\}={\rm
cl}_{\Sigma\mathcal{O}(X)}\\{\bigcup_{f\in\mathcal{F}}f^{-1}(W)\\}\in
H_{c}^{d}(\Sigma\mathcal{O}(X))$

and in $P_{H}(H_{c}^{d}(\Sigma\mathcal{O}(X)))$,

$\overline{\\{{\rm
cl}_{\Sigma\mathcal{O}(X)}\\{f^{-1}(W)\\}:f\in\mathcal{F}\\}}=\overline{\\{{\rm
cl}_{\Sigma\mathcal{O}(X)}\\{\bigcup_{f\in\mathcal{F}}f^{-1}(W)\\}\\}}.$

As $\bigcup_{f\in\mathcal{F}}f^{-1}(W)\in\mathcal{H}$, ${\rm
cl}_{\Sigma\mathcal{O}(X)}\\{\bigcup_{f\in\mathcal{F}}f^{-1}(W)\\}\in\Diamond\mathcal{H}$.
And since $\Diamond\mathcal{H}$ is open in
$P_{H}(H_{c}^{d}(\Sigma\mathcal{O}(X)))$, we have that $\\{{\rm
cl}_{\Sigma\mathcal{O}(X)}\\{f^{-1}(W)\\}:f\in\mathcal{F}\\}\bigcap\Diamond\mathcal{H}\neq\emptyset$.
It follows that there exists $h\in\mathcal{F}$ such that ${\rm
cl}_{\Sigma\mathcal{O}(X)}\\{h^{-1}(W)\\}\in\Diamond\mathcal{H}$. Then ${\rm
cl}_{\Sigma\mathcal{O}(X)}\\{h^{-1}(W)\\}\bigcap\mathcal{H}\neq\emptyset$,
implying that $h^{-1}(W)\in\mathcal{H}$. Therefore, $\mathcal{F}\bigcap
N(\mathcal{H}\leftarrow W)\neq\emptyset$.

Claim 3: ${\rm cl}_{[X\rightarrow Y]_{I}}\mathcal{F}={\rm cl}_{[X\rightarrow
Y]_{I}}\\{g\\}.$

As the specialization order on $[X\rightarrow Y]_{I}$ is the usually pointwise
order on $\mathbb{C}(X,Y)$, ${\rm cl}_{[X\rightarrow
Y]_{I}}\mathcal{F}\subseteq{\rm cl}_{[X\rightarrow Y]_{I}}\\{g\\}$ by the
definition of $g$. On the other hand, by Claim 2 and $\mathcal{F}\in
H([X\rightarrow Y]_{I})\subseteq{\mathsf{Irr}}([X\rightarrow Y]_{I})$, all
basic open sets of $g$ in $[X\rightarrow Y]_{I}$ must meet $\mathcal{F}$. So
$g\in{\rm cl}_{[X\rightarrow Y]_{I}}\mathcal{F}$ or, equivalently, ${\rm
cl}_{[X\rightarrow Y]_{I}}\\{g\\}\subseteq{\rm cl}_{[X\rightarrow
Y]_{I}}\mathcal{F}$. Thus ${\rm cl}_{[X\rightarrow Y]_{I}}\mathcal{F}={\rm
cl}_{[X\rightarrow Y]_{I}}\\{g\\}$.

By Claim 3, the function space $\mathbb{C}(X,Y)$ equipped with the Isbell
topology is an H-sober space. ∎

By Proposition 4.21 and Theorem 4.24, we get the following result.

######  Theorem 4.25.

For $T_{0}$ spaces $X$ and $Y$, the following two conditions are equivalent:

  1. 1.

$Y$ is an H-sober space.

  2. 2.

The function space $\mathbb{C}(X,Y)$ equipped with the Isbell topology is an
H-sober space.

As the $d$-spaces, sober spaces and well-filtered spaces are three special
types of H-sober spaces, by Theorem 4.25, we get the following corollary.

######  Corollary 4.26.

For $T_{0}$ spaces $X$ and $Y$, the following two conditions are equivalent:

  1. 1.

$Y$ is a sober space (resp., $d$-space, well-filtered space).

  2. 2.

The function space $\mathbb{C}(X,Y)$ equipped with the Isbell topology is a
sober space (resp., $d$-space, well-filtered space).

In [10, Corollary 3.12], it was proved that for a core compact space $X$, if
$Y$ is well-filtered, then the function space $\mathbb{C}(X,Y)$ equipped with
the Isbell topology is a well-filtered space. For the well-filteredness,
Corollary 4.26 improves this conclusion by removing the unnecessary condition
that $X$ is core compact.

Finally, we investigate the Scott topology on $\mathbb{C}(X,Y)$.

######  Lemma 4.27.

([3, Lemma II-3.14 and Proposition II-3.15]) Let $X$ be a $T_{0}$ space and
$Y$ a $d$-space. Then

  1. 1.

if $(f_{d})_{d\in D}$ is a net of continuous functions $f_{d}:X\rightarrow Y$
such that $(f_{d}(x))_{d\in D}$ is a directed net of $Y$ (with the
specialization order) for each $x\in X$, then the pointwise sup
$f:X\rightarrow Y$ of the net $f_{d}$ is continuous.

  2. 2.

the subset $\mathbb{C}(X,Y)$ of $(\Omega Y)^{X}$ is closed under directed sups
and hence $\mathbb{C}(X,Y)$ is a dcpo.

######  Corollary 4.28.

For a $T_{0}$ $X$ and a $d$-space $Y$, the function space $\mathbb{C}(X,Y)$
equipped with the Scott topology is a $d$-space.

The following example shows that for sober spaces, Corollary 4.28 does not
hold in general.

######  Example 4.29.

Let $X=1$ be the topological space with only one point and $L$ the complete
lattice constructed by Isbell in [6]. It is well-known that $\Sigma L$ is non-
sober. Let $Y=(L,\upsilon(L))$. Then by [20, Corollary 4.10] or [19,
Proposition 2.9], $Y$ is sober. Clearly, $\mathbb{C}(X,Y)=\\{c_{y}:y\in Y\\}$
and $c_{y}\mapsto y:[X\rightarrow Y]_{\Sigma}\rightarrow\Sigma\Omega Y=\Sigma
L$ is a homeomorphism. So $Y$ is sober but the function space
$\mathbb{C}(X,Y)$ equipped with the Scott topology is non-sober.

It is still not known whether Corollary 4.28 holds for well-filtered spaces.
That is, we have the following question.

######  Question 4.30.

For a $T_{0}$ space $X$ and a well-filtered space $Y$, is the function space
$\mathbb{C}(X,Y)$ equipped with the Scott topology well-filtered?

Let $\mathbb{J}=\mathbb{N}\times(\mathbb{N}\cup\\{\omega\\})$ with ordering
defined by $(j,k)\leq(m,n)$ iff $j=m$ and $k\leq n$, or $n=\omega$ and $k\leq
m$. $\mathbb{J}$ is a well-known dcpo constructed by Johnstone in [7] which is
not sober in its Scott topology.

Figure 1: Johnstone’s dcpo $\mathbb{J}$

######  Proposition 4.31.

There is no well-filtered topology on $\mathbb{J}$ which has the given order
as its specialization order, namely, for any topology
$\upsilon(\mathbb{J})\subseteq\tau\subseteq\sigma(\mathbb{J})$,
$(\mathbb{J},\tau)$ is not well-filtered.

###### Proof.

The set $\mathbb{J}_{max}=\\{(n,\infty):n\in\mathbb{N}\\}$ is the set of all
maximal elements of $\mathbb{J}$ and
$\mathsf{K}(\Sigma~{}\\!\\!\mathbb{J})=(2^{\mathbb{J}_{max}}\setminus\\{\emptyset\\})\bigcup\mathbf{Fin}~{}\\!\mathbb{J}$
(see [11, Example 3.1]). Since
$\upsilon(\mathbb{J})\subseteq\tau\subseteq\sigma(\mathbb{J})$, we have that
$2^{\mathbb{J}_{max}}\setminus\\{\emptyset\\}\subseteq\mathsf{K}((\mathbb{J},\tau))$.
Let $\mathcal{K}=\\{\mathbb{J}_{max}\setminus
F:F\in(\mathbb{J}_{max})^{(<\omega)}\\}$. Then
$\mathcal{K}\subseteq\mathsf{K}((\mathbb{J},\tau))$ is a filtered family and
$\bigcap\mathcal{K}=\bigcap_{F\in(\mathbb{J}_{max})^{(<\omega)}}(\mathbb{J}_{max}\setminus
F)=\mathbb{J}_{max}\setminus\bigcup(\mathbb{J}_{max})^{(<\omega)})=\emptyset$,
but $\mathbb{J}_{max}\setminus F\neq\emptyset$ for all
$F\in(\mathbb{J}_{max})^{(<\omega)}$. Therefore, $(\mathbb{J},\tau)$ is not
well-filtered. ∎

######  Corollary 4.32.

([3, Exercise II-3.16 (V)]) There is no sober topology on $\mathbb{J}$ which
has the given order as its specialization order.

######  Question 4.33.

Is there a well-filtered space $Y$ such that $\Sigma Y$ (i.e., $\Sigma\Omega
Y$) is not well-filtered? Or equivalently, is there a dcpo $P$ and a topology
$\upsilon(P)\subseteq\tau\subseteq\sigma(P)$ such that $(P,\tau)$ is well-
filtered but $(P,\sigma(P))$ is not well-filtered? In particular, is there a
dcpo $P$ such that $(P,\upsilon(P))$ is well-filtered but $(P,\sigma(P))$ is
not well-filtered?

If the answer of Question 4.33 is “Yes”, then the answer of Question 4.30 is
“No”! Indeed, suppose that $Y$ is a well-filtered space for which the Scott
space $\Sigma Y$ is not well-filtered and $X$ is the topological space with
only one point. Then the function space $\mathbb{C}(X,Y)$ equipped with the
Scott topology is not well-filtered since $[X\rightarrow Y]_{\Sigma}$ and
$\Sigma\Omega Y$ are homeomorphic (see Example 4.29).

Conversely, for $\mathcal{R}\geq{\rm H}\geq\mathcal{D}$ (in particular, ${\rm
H}=\mathcal{D},\mathcal{R},\mathsf{WD}$ or $\mathsf{RD}$), the following
example shows that the H-sobriety of function space $\mathbb{C}(X,Y)$ equipped
with the Scott topology does not implies the H-sobriety of $Y$ in general.

######  Example 4.34.

Let $X$ be the topological space with only one point and $[0,1]$ the unit
closed interval with the usual order of reals. Then
$\sigma([0,1])\neq\gamma([0,1])$ (note that $\\{1\\}\in\gamma([0,1])$ but
$\\{1\\}\not\in\sigma([0,1])$). Clearly, the Alexandroff space $\Gamma[0,1]$
is not a $d$-space (since $\gamma([0,1])\nsubseteq\sigma([0,1])$) and the
Scott space $\Sigma[0,1]$ is sober, whence $\Sigma[0,1]$ is H-sober and
$\Gamma[0,1]$ is not H-sober by $\mathcal{R}\geq{\rm H}\geq\mathcal{D}$. As
$[X\rightarrow\Gamma[0,1]]_{\Sigma}$ is homeomorphic to
$\Sigma\Omega(\Gamma[0,1])=\Sigma[0,1]$ (see Example 4.29), the function space
$\mathbb{C}(X,\Gamma[0,1])$ equipped with the Scott topology is H-sober but
$\Gamma[0,1]$ is not H-sober.

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