# Relic jet activity in “Hanny’s Voorwerp” revealed by the LOFAR Two metre Sky
Survey

D. J. B. Smith, M. G. Krause, M. J. Hardcastle and A. B. Drake
Centre for Astrophysics Research, Department of Physics, Astronomy and
Mathematics, University of Hertfordshire, Hatfield AL10 9AB, UK E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We report new observations of “Hanny’s Voorwerp” (hereafter HV) taken from the
second data release of the LOFAR Two-metre Sky Survey (LoTSS). HV is a highly-
ionised region in the environs of the galaxy IC2497, first discovered by the
Galaxy Zoo project. The new 150 MHz observations are considered in the context
of existing multi-frequency radio data and archival narrow-band imaging from
the Hubble Space Telescope, centred on the [Oiii] emission line. The combined
sensitivity and spatial resolution of the LoTSS data – which far exceed what
was previously available at radio frequencies – reveal clear evidence for
large-scale extended emission emanating from the nucleus of IC2497. The radio
jet appears to have punched a hole in the neutral gas halo, in a region co-
located with HV. The new 150 MHz data, alongside newly-processed archival 1.64
GHz eVLA data, reveal that the extended emission has a steep spectrum,
implying an age $>10^{8}$ yr. The jet supplying the extended 150 MHz structure
must have “turned off” long before the change in X-ray luminosity reported in
recent works. In this picture, a combination of jet activity and the influence
of the radiatively efficient active galactic nucleus are responsible for the
unusual appearance of HV.

###### keywords:

Galaxies: active – Galaxies: jets – Galaxies: peculiar

††pubyear: 2022††pagerange: Relic jet activity in “Hanny’s Voorwerp” revealed
by the LOFAR Two metre Sky Survey–Relic jet activity in “Hanny’s Voorwerp”
revealed by the LOFAR Two metre Sky Survey

##  1 Introduction

Hanny’s Voorwerp (hereafter HV) is a region of highly ionised material tens of
kpc in projected size, which is located near the $z=0.05$ galaxy IC2497. HV
was originally reported in Lintott et al. (2009), having been discovered
during visual classification of galaxies in the Sloan Digital Sky Survey
(SDSS; York et al., 2000), as part of the Galaxy Zoo project (Lintott et al.,
2008). HV was identified as a result of its unusual morphology and extremely
bright [Oiii] emission falling in the SDSS g’ band filter. Lintott et al.
(2009) suggested that HV could be explained either as a highly photoionised
region resulting from an active galactic nucleus (AGN) with an unusual dust
geometry that prevents it from ionising the host galaxy’s own nuclear gas, or
as a “light echo" resulting from a dramatic change in the luminosity of the
central source over the past $10^{5}$ years.

As well as discovering a halo containing $\sim 10^{9}$ M⊙ of HI gas
surrounding IC2497 and HV, Józsa et al. (2009, hereafter J09) provided the
first evidence of jets visible as a marginal extension to an already
elliptical restoring beam in both 1.4 and 4.9 GHz observations in the
direction of HV. This extension is not visible in 1.4 GHz observations from
the Faint Images of the Radio Sky at Twenty-centimetres survey (FIRST; Becker
et al., 1995), the NRAO VLA Sky Survey (NVSS; Condon et al., 1998), the
Westerbork Northern Sky Survey (WENSS; Rengelink et al., 1997), or in 150 MHz
observations from the TIFR GMRT Sky Survey Alternative Data Release (TGSS-ADR;
Intema et al., 2017). Nevertheless, Rampadarath et al. (2010) further
underlined the presence of jets using MERLIN observations to detect two
components separated by $\sim$300 pc with brightness temperatures in excess of
$10^{5}$ K (an upper limit to the brightness temperature that can be
attributed to star forming regions; Condon et al., 1991; Biggs et al., 2010),
though with a significant flux deficit relative to the integrated measurements
reported by J09 indicating the presence of a resolved nuclear starburst in
IC2497.

Hubble Space Telescope observations in both broad- and narrow-band filters
centred on the [Oiii] and H$\alpha$+[Nii] emission lines (Keel et al., 2012a)
found evidence for large regions photoionised by AGN activity, with evidence
for some sites within HV dominated by star formation. Keel et al. (2012a) also
hypothesised about the role of a previous major merger in stirring up the Hi
gas around IC2497, and producing the bar and strong warping visible in the
disk of IC2497.

Several works have looked at HV at X-ray wavelengths, including Schawinski et
al. (2010) and Sartori et al. (2018). These works show that HV contains a
Compton-thick AGN which has recently (in the last $\sim$70,000 yr) undergone a
dramatic change in luminosity, similar to the change suggested by Keel et al.
(2012b) and Lintott et al. (2009). Most recently, Fabbiano & Elvis (2019)
found evidence for extended soft X-ray emission in Chandra data of HV,
spatially consistent (albeit at low-resolution and with low statistical
significance) with the direction of the extension similar to that in the radio
data observed by J09.

Our observational capabilities at radio frequencies have exploded since HV was
first studied with interferometry by J09 and Rampadarath et al. (2010). The
capabilities of the Low Frequency Array (LOFAR; van Haarlem et al., 2013)
exemplify the huge steps in survey speed, sensitivity, resolution and imaging
capabilities. One of the key drivers for LOFAR since its inception has been to
conduct surveys of the whole northern sky, and the LOFAR Two-metre Sky Survey
(LoTSS Shimwell et al., 2017) is well on the way to fulfilling that aim. The
first data release (LoTSS DR1; Duncan et al., 2019; Shimwell et al., 2019;
Williams et al., 2019) covered 424 deg2 over the HETDEX (Hill et al., 2008)
spring field using the LOFAR High Band Antenna (HBA) at a central frequency of
150 MHz with a mean RMS sensitivity $<100\,\mu$Jy and resolution around 6
arcsec. As well as the advantages of a huge increase in sensitivity relative
to FIRST, LoTSS HBA data benefit from including short and long-baseline
observations in the pipeline-processed data products, meaning that it is
uniquely sensitive to low-frequency emission on both compact and extended
scales. LoTSS also includes even deeper observations over tens of deg2 in the
prime Northern fields with the best ancillary data (Boötes, Lockman Hole and
ELAIS-N1) in the first data release of the LoTSS Deep fields (Tasse et al.,
2021; Sabater et al., 2021; Kondapally et al., 2021; Duncan et al., 2021).

Progress continues to be rapid; as well as observations at even lower
frequencies (using the Low Band Antenna; de Gasperin et al., 2021) and using
international baselines to obtain sub-arsecond resolution (Morabito et al.,
2022), the second data release of LoTSS (DR2; Shimwell et al., 2022) reaches a
median 150 MHz sensitivity of 84 $\mu$Jy beam-1 over $5700$ deg2, detects more
than 4.3 million 150 MHz sources, and benefits from a range of further
improvements to the data processing relative to LoTSS DR1.

In addition to the sensitivity of the LoTSS maps, together with higher
frequency data, 150 MHz observations are particularly useful for determining
the properties of extragalactic radio sources. Energy loss in a population of
relativistic electrons is expected to vary as a function of $\nu^{2}$, such
that higher frequencies fade more rapidly as the electron population ages, and
as a result the degree of radio spectral curvature increases with time. When
studied in the context provided by GHz data therefore, low frequency
observations can ‘anchor’ the spectrum, and in doing so enable the best
measurements of spectral curvature (and therefore the age of the electron
population; e.g. Kardashev, 1962; Laing & Peacock, 1980; Harwood et al.,
2013). Using data spanning $325\,\mathrm{MHz}<\nu<4.9\,$GHz, J09 reported a
flat radio spectrum of the core of IC2497, consistent with a power law such
that $S_{\nu}\propto\nu^{-\alpha}$ with an index of $\alpha=0.55$, and no
evidence for spectral curvature indicating that the core radio source is
young.

The IC2497-HV system is thought to be virtually unique in the local Universe
(Keel et al., 2012b, 2015). Even in the absence of HV, the unambiguous
presence of jets in IC2497 alone would make the system remarkable since Singh
et al. (2015) identified only four instances of double radio jets among
187,000 spiral galaxies identified in SDSS imaging (see also Mao et al., 2015;
Nesvadba et al., 2021). HV therefore represents a prototype for studying a
broad range of phenomena: the variability thought to be inherent in AGN
physics, the role of mergers and/or interactions in triggering AGN activity,
as well as the influence of AGN activity on the surrounding gas. These are
some of the key physical processes required to explain the observed properties
of galaxies (see e.g. Silk & Rees, 1998; Croton et al., 2006; Alexander &
Hickox, 2012). In this paper, we examine the properties of IC2497 and HV in
the new 150 MHz data from LoTSS DR2, and in newly processed archival 1.6 GHz
eVLA data. In Section 2 we describe the data we use, while in Section 3 we
present our results before making some concluding remarks in Section 4.

##  2 Data

New LoTSS DR2 observations of HV are available from the LOFAR surveys
website111www.lofar-surveys.org. The properties of the LoTSS DR2 mosaics are
described in detail in Shimwell et al. (2022), however the vital statistics
can be summarised as follows. The 150 MHz images are automatically reduced
using the latest version of the LoTSS processing pipeline, which uses
direction-dependent methods to account for varying ionospheric conditions,
giving high quality data with 6-arcsec resolution, and sensitivity to emission
on both small and large scales due to the range of baselines available to the
Dutch LOFAR array. HV falls within the area covered by the P144+35 mosaic;
however, to obtain an optimal image of the region surrounding HV, we produced
a self-calibrated image following the method of van Weeren et al. (2021). The
resulting image, shown in the left panel of Figure 1, is at the natural
resolution of the LOFAR observations using robustness 0.5, and has an
elliptical restoring beam with major and minor axes 8.5 and 4.8 arcsec, 1.5
arcsec pixels and a uniform RMS noise level of 92 $\mu$Jy beam-1 across the
image. The absolute flux scale of the LoTSS data is thought to be correct at
the level of 6 percent.

Figure 1: New LoTSS (left) and eVLA (right) images of the region surrounding
IC2497 and HV, highlighted with the blue circle, which has a radius of 1
arcmin. The size of the natural-resolution restoring beam is shown as the
black ellipse within a box in the lower-left corner of each image. The flux
scale is as indicated by the colour bars to the right.

We obtained archival eVLA data of the region surrounding HV, taken in 2011 in
B-configuration (giving excellent sensitivity to extended structures out to
angular scales of 2 arcmin). The eVLA data have an effective frequency of 1.64
GHz, with 256 MHz of bandwidth, and provide similar spatial resolution (major
and minor beam axes of 4.9 and 3.2 arcsec) to the new LoTSS data. The newly-
reduced eVLA map is shown in the right-hand panel of figure 1; based on
comparing the flux densities of unresolved sources with FIRST, we conclude
that the flux scale is correct to $\sim 5$ percent. We also obtained the 1.4
GHz WSRT image and Hi data cube from J09 (G.I.G. Józsa, private
communication).

The HV-IC2497 system was observed using Advanced Camera for Surveys (ACS)
tunable ramp filters on board the HST (Keel et al., 2012a), with the central
wavelengths set to sample the redshifted [Oiii] and H$\alpha$ emission lines,
and integration times of 2570 s and 2750 s respectively. The pointing was
chosen so that HV and IC2497 fell within the monochromatic field of view (40
arcsec $\times$ 80 arcsec) and no continuum subtraction was attempted since
Lintott et al. (2009) showed that it was unnecessary. The final reduced HST
data products from Keel et al. (2012a) were provided by the authors (W.C.
Keel, private communication).

##  3 Results

###  3.1 Morphology and photometry

Figure 2 shows the new LoTSS 150 MHz mosaic as contours overlaid on a colour
composite image derived using the HST [Oiii] and H$\alpha$ narrow-band images
from Keel et al. (2012a). It is clear that as well as an unresolved component
coincident with the centre of IC2497, the extension identified by J09 is now
resolved into clear jet-related emission spatially coincident with HV, and
with a flux density of $10.0\pm 0.1$ mJy. The total flux density of the system
at 150 MHz in the new LoTSS mosaic is $97.3\pm 0.2$ mJy. This structure is not
detected in the eVLA image (or any of the other assembled radio data),
implying that its spectrum is steep.

Figure 2: LoTSS 150 MHz map of the region centred on HV (white contours)
overlaid on a colour composite built from the [Oiii] (green channel) and
H$\alpha$ (blue and red channel) narrow-band filter HST images from Keel et
al. (2012a). The LoTSS contours, shown in magenta, are at levels of 0.2, 0.5,
1, 2, 5, 10, 20 & 50 mJy beam-1.

The flux density of the unresolved core of IC2497 (obtained by subtracting
$S_{\nu}^{\mathrm{Extended}}$ from $S_{\nu}^{\mathrm{Total}}$) is consistent
with the measurement from the TGSS-ADR mosaic once the calibration
uncertainties are taken into account, and corresponds to a 150 MHz luminosity
of $(5.63\pm 0.01)\times 10^{23}$ W Hz-1. Assuming a standard relationship
from Heckman & Best (2014) and correcting to 150 MHz following Sabater et al.
(2019), this enables us to estimate a mechanical power in the jet of
$1.5\times 10^{36}$ W. Although it compares well with the Lintott et al.
(2009) estimate of HV’s [Oiii] luminosity ($1.5\times 10^{35}$ W), the
energetics alone preclude the current core activity from making the dominant
contribution to the Keel et al. (2012a) estimate of the luminosity required to
maintain the ionization of HV ($\sim 10^{38}$ W).

Using the 100 $\mu$m flux density quoted for IC2497 in the IRAS faint source
catalogue (Moshir et al., 1990), we can estimate the star formation rate (SFR)
assuming that the dust is heated solely by young stars, that it has a far-
infrared SED similar to M82 (Polletta et al., 2007) and the canonical
relationship between far-infrared luminosity and SFR from Kennicutt (1998)
adjusted to our adopted initial mass function (IMF) from Chabrier (2003).
Doing so, we obtain a far infrared luminosity integrated between 8-1000 $\mu$m
of $4.1\times 10^{11}L_{\odot}$ – putting IC2497 in the Luminous Infrared
Galaxy (LIRG) class – and an SFR of $\sim 40\,$M⊙ yr-1, although we note that
the uncertainties inherent in our choice of SED template and IMF (and
therefore the derived SFR) are significant. Nevertheless, comparing the
derived SFR with the mass-independent $\mathrm{SFR}-L_{\mathrm{150\,MHz}}$
relations from Gürkan et al. (2018) and Smith et al. (2021), reveals no
evidence for any radio excess, consistent with the nuclear starburst reported
by Rampadarath et al. (2010). Nevertheless, the morphological evidence for the
presence of an AGN is conclusive.

Frequency |  $S_{\nu}^{\mathrm{Total}}$ (mJy) |  $S_{\nu}^{\mathrm{Extended}}$ (mJy) | Facility/Survey | Reference
---|---|---|---|---
150 MHz { | $97.3\pm 0.2$ | $10.0\pm 0.1$ | LoTSS | This work
$66.4\pm 8.3$ |  | TGSS-ADR | Intema et al. (2017)
325 MHz | $51\pm 4.9$ |  | WENSS | Rengelink et al. (1997)
1.4 GHz { | $20.9\pm 1.1$ | $3.2\pm 0.2$ | WSRT | Józsa et al. (2009)
$18.5\pm 0.6$ |  | NVSS | Condon et al. (1998)
$16.8\pm 0.9$ |  | FIRST | Becker et al. (1995)
1.64 GHz | $14.2\pm 0.5$ | $0.48\pm 0.29$ | eVLA | This work
4.9 GHz | $11.6\pm 0.6$ |  | WSRT | Józsa et al. (2009)
Table 1: A compilation of radio frequency flux densities of IC2497 & HV,
including both the total ($S_{\nu}^{\mathrm{Total}}$) and extended
($S_{\nu}^{\mathrm{Extended}}$) components where available.

To put the new morphological information available from the $\sim$6 arcsec
resolution LoTSS data in the context of the previously available radio data,
in both panels of Figure 3 we overlay the 150 MHz data as magenta contours on
a background image showing the Hi data from J09. Overlaid in the left panel is
the 1.4 GHz WSRT image from J09, shown as blue contours (the WSRT data have an
elliptical point spread function of $14\times 11$ arcsec, shown as the blue-
filled ellipse in the lower-left corner; Morganti et al., 2004), with the
details of the chosen contour levels given in the caption. While at 1.4 GHz
the degree of extension in the radio source is perhaps debatable on the basis
of visual inspection alone due to the elongated beam along the IC2497-HV
direction, this is not true at 150 MHz, where the LoTSS PSF (shown as the
magenta ellipse to the lower-left) is more than four times smaller. In the
right panel of Figure 3 we show a zoomed version of the central region; for
the first time it is clear that the clearly-resolved structure emanating from
the nucleus of IC2497 coincides with a minimum in the surrounding
$10^{9}\,M_{\odot}$ reservoir of Hi gas found by J09 (shown in the background
image).

Based on the left panel of Figure 3 it would be tempting to identify the knot
of coincident 150 MHz and 1.4 GHz emission on the Northern side of IC2497 as
being related to the counter-jet, emanating in the direction opposite to HV.
However, inspection of the SDSS images shows that this knot of emission is
instead associated with an edge-on late-type galaxy interloper.

Figure 3: Composite image showing the Hi image (background, greyscale) and the
new LoTSS data (magenta contours, levels of 0.2, 0.5, 1, 2, 5, 10, 20 & 50 mJy
beam-1). In the left panel, the WSRT 1.4 GHz image from Józsa et al. (2009) is
shown as blue contours (with contour levels of 0.1, 0.2, 0.5, 1, 2, 5, 10 & 20
mJy beam-1), and the filled ellipses in the box to the bottom left indicate
the size of the restoring beam, colour-coded to match the contours. The right
panel is zoomed in to highlight the spatial anti-correlation between the jet
(magenta contours) and the Hi gas (background greyscale) with the same contour
levels and colour scale as the left panel.

###  3.2 Radio spectral index

To examine the radio spectrum of the IC2497-HV system, in Figure 4 we show the
total photometry ($S_{\nu}^{\mathrm{Total}}$) from Table 1 as filled circles,
though we have added calibration uncertainties in quadrature, at the level of
6 percent to the new LoTSS flux density (Shimwell et al., 2022), and 5 percent
to the new eVLA measurements. The photometry is overlaid with a power law with
the spectral index $\alpha=0.55$ found by J09. Despite reaching lower
frequencies, the new LoTSS data do not reveal any evidence for spectral
curvature in the core; the power law found by J09 continues down to
frequencies of 150 MHz at least.

In figure 4, the extended source photometry ($S_{\nu}^{\mathrm{Extended}}$
from Table 1) is shown using square symbols. There are conflicting results for
the spectral index of the extended component, depending on whether we use the
WSRT or eVLA flux densities, since the nominal 1.64 GHz eVLA flux density of
$0.48\pm 0.29$ mJy (measured at a spatial resolution similar to the new LoTSS
data) is inconsistent with the 1.4 GHz WSRT estimate of $3.2\pm 0.2$ mJy. We
suggest that the flat spectrum of the extended emission
($\alpha_{150}^{1400}=0.51\pm 0.03$) obtained using the WSRT estimated flux
density may be an artefact of contamination from the extension in the North-
South direction of the WRST point spread function. Using instead the eVLA flux
density alongside the 150 MHz measurement we obtain our best estimate of the
extended component’s spectral index of
$\alpha_{150}^{1640}=1.25^{+0.28}_{-0.23}$. As the extended structure detected
in the LOFAR data is not formally detected in the eVLA image, the spectrum
might be even steeper, but we can be confident that $\alpha>1$.

The steep spectrum implies that the plasma in the extended structure is
significantly older than that in the core, as expected if the jet feeding it
has ‘turned off’ some time ago. However, since we have measurements at just
two frequencies, and a detection only at 150 MHz, detailed analysis is not
possible. If we assume that the injection index for this material is
$\alpha=0.55$, estimate a self-consistent equipartition magnetic field
strength of around 0.2 nT (based on treating the extended material as a
uniformly filled sphere of radius 16 arcsec), and take account of inverse-
Compton losses to the CMB at the redshift of HV222Calculations were done using
the pysynch Python package https://github.com/mhardcastle/pysynch which
provides an interface to the synchrotron emission code of Hardcastle et al.
(1998)., then using a Jaffe-Perola model for spectral ageing (Jaffe & Perola,
1973) we find that the spectral age must be $\ga 10^{8}$ yr to give
$\alpha>1$. Adiabatic expansion of a lobe generated by a now disconnected jet
will tend to increase the apparent spectral age, perhaps by an order of
magnitude, but this still implies that the jet supplying the extended
structure must have turned off long before the change in X-ray luminosity
reported by Sartori et al. (2018). Future 60 MHz observations with the LOFAR
Low-band Antennae may be able to provide better constraints on the degree of
spectral curvature, and therefore on the age of the plasma.

Figure 4: Radio frequency spectra of HV, including values for the total (blue
circles) and extended components (red squares), as detailed in Table 1. The
solid blue line indicates a power-law spectrum with spectral index
$\alpha=-0.55$, while the dotted and dot-dashed lines indicate the spectral
index obtained using the 150 MHz data alongside the WSRT 1.4 GHz or eVLA 1.64
GHz flux densities, which are formally inconsistent as discussed in the text.

##  4 Discussion & Conclusions

We attempt to bring all of this information together as follows. The flat
spectral index in the core of IC2497 extends to 150 MHz, and is similar to
values for injection indices assumed by jet models (see e.g. Shulevski et al.,
2015, for a discussion). This is consistent with the small-scale nuclear jets
observed by Rampadarath et al. (2010).

On the other hand, the extended component seen at 150 MHz has a steep spectrum
($\alpha_{150}^{1640}=1.25^{+0.28}_{-0.23}$), implying an age of order
$10^{8}$ yr if the injection index is the same as that measured for the
compact component. Other studies based at X-ray wavelengths (e.g. Schawinski
et al., 2010; Sartori et al., 2018) have suggested a timescale of $\sim
10^{5}$ years for a change in the luminosity of the central engine; however
the LoTSS data conclusively show that the jet that generated the extended
emission must have been unrelated to that nuclear outburst.

The brightest knots of 150 MHz emission visible in Figure 2 appear co-located
with the main star-forming structure at the Northern end of HV, and with the
apparently separate (in the HST data) component to the West. The Northern 150
MHz knot appears coincident with the filamentary “fingers" located just to the
South of the star-forming region located at the Northern tip of HV (which
appears pink in figure 2 due to the bright H$\alpha$ emission). These features
were first identified in Keel et al. (2012a) as possible evidence of gas
entrained by ram pressure of the jet emanating from the core of IC2497.
Radiative dissipation in turbulent mixing may contribute to the line emission
locally (Krause & Alexander, 2007), even though the overall spectrum is
dominated by photoionisation (compare below).

The “hole" feature on the South-East side of HV is reminiscent of supernova
remnants visible within our own galaxy, although the overall structure and the
optical line ratios in this area measured by Lintott et al. (2009, based on
long-slit spectroscopy) and Keel et al. (2012a, based on HST narrow-band
imaging) are clearly dominated by photoionisation, and therefore not
associated with the passage of the jet. Fabbiano & Elvis (2019) reported
extended X-ray emission apparent in this area (albeit at low statistical
significance) as might also be expected under the hypothesis of a recent
explosive event at this location, although no remnant is visible in any of the
assembled data.

In the Krause (2005) model, the leading shell of a starburst wind (perhaps
resulting from the nuclear starburst suggested to be present in the HV system
by Rampadarath et al., 2010) can cool and form a dense, mainly neutral Hi
shell, if for some reason it encounters enough gas, such as in the post-major
merger scenario for IC2497 discussed by Keel et al. (2012a). When a large-
scale jet is also present (as is clearly the case based on the new 150 MHz
data), it interacts with the shell and pierces a hole in it. In the Krause
(2005) simulations, the hole is $\sim$20 kpc in diameter, very similar to the
radio lobe width observed in HV.

A counterargument to this picture appears to be that in the Lintott et al.
(2009) optical spectroscopy, as well as in unpublished integral-field
spectroscopy of the HV system using the WIYN-HEXPAK instrument (W.C. Keel,
private communication) HV has a smooth velocity field showing no evidence of
disturbances that might be expected in a jet interaction scenario (see Krause
2005, Fig. 3). While a blue-shifted velocity field that is coherent over many
kpc might indeed point to a starburst wind shell, the scenario from Krause
(2005) could still describe HV but must be modified. An element of physics
that was missing in the latter simulations was the Vishniac shell instability
(Vishniac, 1983): 3D high-resolution simulations have shown that such shells
fragment into filaments and clumps (e.g. van Marle & Keppens, 2012; Krause et
al., 2013). The disturbed kinematics in the Krause (2005) simulations come
from the strong early resistance of the unfragmented shell with high pressure
build-up inside of it and strong gas acceleration when the shell eventually
fragments due to other instabilities. If the shell in reality fragments
earlier, the jet might even better pierce the immediate impact region between
the clumps with less effect on the surrounding shell. This could be tested
with dedicated simulations.

Alternatively, the spatial coincidence between jet and Hi minimum could be
purely a projection effect (in which case the Hi minimum may be apparent due
to the gas at this location being almost fully ionised), or that the jet
punched through the gas sufficiently long ago that any turbulence has subsided
(e.g. Krumholz et al., 2006). If we assume that the timescale for turbulent
decay is similar to the crossing time of the emission-line region, back of the
envelope calculations (dividing the size of the region by the assumed speed,
adopting an initial turbulent velocity of 100 km s-1, similar to velocities
found e.g. from the simulation of Krause 2005 or the observations of Nesvadba
et al. 2021, and a region size of 10 kpc) produce timescales of the order of
$10^{8}$ yr, which is comparable to the lower limit on the lobe age estimated
in section 3.2. We would therefore expect any turbulent motions of this
magnitude driven into the emission-line region by the expanding lobes not to
be visible by the present time.

The picture that emerges from the assembled data is one with multiple events
over different timescales. First, a tidal encounter left a $10^{9}$ M⊙ cloud
of Hi around IC2497, and caused IC2497’s warped appearance. The new radio
frequency data then show that a radio outburst, $\sim 10^{8}$ years ago,
shaped and perturbed the multi-phase gas tens of kpc around IC2497, before
switching off. The relic radio lobe from this outburst is what we now see in
the LOFAR data – a corresponding northern radio lobe has presumably faded to
invisibility, perhaps because of the lower gas density to the north of IC2497.
Much more recently ($\sim 10^{5}$ years ago) a radiatively-efficient AGN
outburst illuminated the gas that had been swept up around the radio lobe and
gave rise to the extended emission-line region that characterizes HV. The
ongoing nuclear starburst and jet (both first observed by Rampadarath et al.,
2010) may have begun/turned on around the same time but it seems unlikely that
the jet was continuously active through the period. This timeline is clearly
in contrast with the previous suggestion that fading AGN could indicate a
lasting change in accretion mode (from being dominated by radiative to
mechanical output; e.g. Sartori et al., 2018). In the IC2497-HV system, the
situation is clearly more complex, with evidence for recurring episodes of
mechanical (or radiatively inefficient) jet activity in addition to the
radiatively efficient activity responsible for ionising HV.

We echo previous suggestions that HV could be a unique local object that
exemplifies processes much more common in the high-redshift Universe at the
peak of cosmic star formation, and is therefore worthy of more study.

## Acknowledgements

DJBS dedicates this work to the loving memory of D.H.F. Smith (1943-2021):
father, Chieftain, poet, and Voorwerp enthusiast. The authors would like to
thank W.C. Keel for refereeing this paper, for providing the fully-processed
HST tunable ramp filter data from Keel et al. (2012a), and for a preview of
WIYN-HEXPAK observations of HV. We also thank G.I.G. Jósza for providing the
FITS data products from Józsa et al. (2009). The authors would like to thank
Paul Haskell and Soumyadeep Das for valuable comments. DJBS and MJH
acknowledge support from the UK Science and Technology Facilities Council
(STFC) under grant ST/V000624/1. LOFAR is the Low Frequency Array designed and
constructed by ASTRON. It has observing, data processing, and data storage
facilities in several countries, which are owned by various parties (each with
their own funding sources), and that are collectively operated by the ILT
foundation under a joint scientific policy. The ILT resources have benefited
from the following recent major funding sources: CNRS-INSU, Observatoire de
Paris and Université d’Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science
Foundation Ireland (SFI), Department of Business, Enterprise and Innovation
(DBEI), Ireland; NWO, The Netherlands; The Science and Technology Facilities
Council, UK; Ministry of Science and Higher Education, Poland; The Istituto
Nazionale di Astrofisica (INAF), Italy.

This research made use of the Dutch national e-infrastructure with support of
the SURF Cooperative (e-infra 180169) and the LOFAR e-infra group. The Jülich
LOFAR Long Term Archive and the German LOFAR network are both coordinated and
operated by the Jülich Supercomputing Centre (JSC), and computing resources on
the supercomputer JUWELS at JSC were provided by the Gauss Centre for
Supercomputing e.V. (grant CHTB00) through the John von Neumann Institute for
Computing (NIC).

This research made use of the University of Hertfordshire high-performance
computing facility and the LOFAR-UK computing facility located at the
University of Hertfordshire and supported by STFC [ST/V002414/1], and of the
Italian LOFAR IT computing infrastructure supported and operated by INAF, and
by the Physics Department of Turin University (under an agreement with
Consorzio Interuniversitario per la Fisica Spaziale) at the C3S Supercomputing
Centre, Italy.

## Data Availability

The new LoTSS 150 MHz observations presented in this work are available from
the LOFAR Surveys website, https://www.lofar-surveys.org/ as part of the
second data release of the LOFAR Two-metre Sky Survey (LoTSS).

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# Further Studies on Open Well-filtered Spaces

Chong Shen  Xiaoyong Xi  Dongsheng Zhao  School of Science
Beijing University of Posts and Telecommunications
Beijing, China  School of Mathematics and Statistics
Yancheng Terchers University
Yancheng, Jiangsu Province, China  Mathematics and Mathematics Education,
National Institute of Education
Nanyang Technological University
1 Nanyang Walk, Singapore

###### Abstract

The open well-filtered spaces were introduced by Shen, Xi, Xu and Zhao to
answer the problem whether every core-compact well-filtered space is sober. In
the current paper we explore further properties of open well-filtered spaces.
One of the main results is that if a space is open well-filtered, then so is
its upper space (the set of all nonempty saturated compact subsets equipped
with the upper Vietoris topology). Some other properties on open well-filtered
spaces are also studied.

###### keywords:

well-filtered space, sober space, core-compact space, locally compact space,
open well-filtered space

††journal: Electronic Notes in Theoretical Informatics and Computer
Science††volume: 2††thanks: This work was supported by the National Natural
Science Foundation of China (1210010153, 12071188) and Jiangsu Provincial
Department of Education (21KJB110008).††thanks: Email:
<EMAIL_ADDRESS>Email<EMAIL_ADDRESS>Email:
<EMAIL_ADDRESS>
##  1 Introduction

The open well-filtered spaces were introduced in [6] and used to give an
alternative and more natural proof of the conjectured conclusion that every
core compact and well-filtered space is sober. In [6], the authors actually
proved a stronger conclusion: every core-compact and open well-filtered space
is sober, as every well-filtered space is open well-filtered and the converse
conclusion is not true in general. By [8] and [10], one knows that a space is
well-filtered if and only if its upper space is well-filtered. It is thus
natural to wonder whether a similar result holds for open well-filteredness.
In this paper we shall give a partial answer to this problem and prove that if
a space is open well-filtered then so is its upper space. A number of new
results on open well-filtered spaces will also be presented here.

##  2 Preliminaries

We first recall some basic definitions and results that will be used in the
paper.

Let $P$ be a poset. A nonempty subset $D$ of $P$ is _directed_ (resp.,
_filtered_) if every two elements of $D$ have an upper (resp., lower) bound in
$D$. A poset $P$ is a _directed complete poset_ , or a _dcpo_ for short, if
for any directed subset $D\subseteq P$, the supremum $\bigvee D$ exists.

For any subset $A$ of a poset $P$,

$\mathord{\uparrow}A=\\{y\in P:\exists x\in A,x\leq y\\},{\mbox{ and
}}\mathord{\downarrow}A=\\{y\in P:\exists x\in A,y\leq x\\}.$

In particular, for each $x\in X$, we write
$\mathord{\uparrow}x=\mathord{\uparrow}\\{x\\}$ and
$\mathord{\downarrow}x=\mathord{\downarrow}\\{x\\}$.

For $x,y\in P$, $x$ is _way-below_ $y$, denoted by $x\ll y$, if for any
directed subset $D$ of $P$ with $\bigvee D$ existing, $y\leq\bigvee D$ implies
$x\leq d$ for some $d\in D$. Denote
$\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadrightarrow$}}x=\\{y\in P:x\ll
y\\}$ and
$\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadleftarrow$}}x=\\{y\in P:y\ll
x\\}$. A poset $P$ is _continuous_ , if for any $x\in P$, the set
$\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadleftarrow$}}x$ is directed and
$x=\bigvee\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadleftarrow$}}x$. A
continuous dcpo is also called a _domain_.

A subset $U$ of a poset $P$ is _Scott open_ if (i) $U=\mathord{\uparrow}U$ and
(ii) for any directed subset $D$ of $P$ with $\bigvee D$ existing, $\bigvee
D\in U$ implies $D\cap U\neq\emptyset$. All Scott open subsets of $P$ form a
topology on $P$, called the _Scott topology_ on $P$ and denote by $\sigma(P)$.
The space $\Sigma P=(P,\sigma(P))$ is called the _Scott space_ of $P$.

For a $T_{0}$ space $X$, the specialization order $\leq$ on $X$ is defined by
$x\leq y$ iff $x\in{\rm cl}_{X}(\\{y\\})$, where ${\rm cl}_{X}$ is the closure
operator of $X$. Clearly, ${\rm cl}_{X}(\\{y\\})=\mathord{\downarrow}y$.

In what follows, if no otherwise specified, the partial order on a $T_{0}$
space will mean the specialization order.

######  Remark 2.1.

[1, 2] For each poset $(P,\leq_{P})$, the specialization order on $\Sigma P$
coincides with $\leq_{P}$.

For any $T_{0}$ space $X$, we shall often use $\mathcal{O}(X)$ to denote the
topology of $X$ (the collection of all open sets of $X$). For any subset $A$
of $X$, the _saturation_ of $A$, denoted by $Sat_{X}(A)$, is defined by

$Sat_{X}(A)=\bigcap\\{U\in\mathcal{O}(X):A\subseteq U\\},$

or equivalently, $Sat_{X}(A)=\mathord{\uparrow}A$ with respect to the
specialization order (see [2, Proposition 4.2.9]). A subset $A$ of $X$ is
_saturated_ if $A=Sat_{X}(A)$.

For any $U,V\in\mathcal{O}(X)$, we write $U\ll V$ for that $U$ is way-below
$V$ in the poset $(\mathcal{O}(X),\subseteq)$. Using a similar proof to that
of the Alexander’s Subbase Lemma (see [2, Theorem 4.4.29]), we can obtain the
following result.

######  Lemma 2.2.

Let $X$ be a $T_{0}$ space, $\mathcal{S}$ be a subbase for $\mathcal{O}(X)$,
and $U,V\in\mathcal{O}(X)$. Then $U\ll V$ if and only if one can extract a
finite subcover of $U$ from any cover $\\{U_{i}:i\in I\\}\subseteq\mathcal{S}$
of $V$.

######  Definition 2.3.

[6] Let $X$ be a $T_{0}$ space.

  1. (1)

A subfamily $\mathcal{F}\subseteq\mathcal{O}(X)$ is called _$\ll$ -filtered_,
denoted by $\mathcal{F}\subseteq_{flt}\mathcal{O}(X)$, if for any
$U_{1},U_{2}\in\mathcal{F}$, there exists $U_{3}\in\mathcal{F}$ such that
$U_{3}\ll U_{1}$ and $U_{3}\ll U_{2}$.

  2. (2)

$X$ is called _open well-filtered_ if for each $\ll$-filtered family
$\mathcal{F}\subseteq\mathcal{O}(X)$ and $U\in\mathcal{O}(X)$,
$\bigcap\mathcal{F}\subseteq U$ implies that $V\subseteq U$ for some
$V\in\mathcal{F}.$

######  Proposition 2.4.

[6] If $X$ is an open well-filtered space and $\\{U_{i}:i\in I\\}$ is a
$\ll$-filtered family of nonempty open sets, then $\bigcap\\{U_{i}:i\in I\\}$
is a nonempty compact saturated set.

######  Definition 2.5.

[1, 2] A $T_{0}$ space $X$ is called _core-compact_ if for each $x\in X$ and
$U\in\mathcal{O}(X)$ such that $x\in U$, there exists $V\in\mathcal{O}(X)$
such that $x\in V\ll U$.

######  Remark 2.6.

[1, 2] A $T_{0}$ space $X$ is core-compact if and only if the poset
$(\mathcal{O}(X),\subseteq)$ is continuous.

######  Theorem 2.7.

[6] Every core-compact open well-filtered space is sober.

######  Definition 2.8.

[1, 2] A $T_{0}$ space $X$ is called _well-filtered_ if for any filtered
family $\\{Q_{i}:i\in I\\}$ of compact saturated subsets of $X$ and any open
set $U\subseteq X$, $\bigcap_{i\in I}Q_{i}\subseteq U$ implies
$Q_{i_{0}}\subseteq U$ for some $i_{0}\in I$.

######  Definition 2.9.

[1, 2] A nonempty subset $A$ of a topological space $X$ is called
_irreducible_ if for any closed sets $F_{1}$, $F_{2}$ of $X$, $A\subseteq
F_{1}\cup F_{2}$ implies $A\subseteq F_{1}$ or $A\subseteq F_{2}$. A $T_{0}$
space $X$ is called _sober_ , if for any irreducible closed set $F$ of $X$,
there is a (unique) point $x\in X$ such that $F={\rm cl}_{X}(\\{x\\})$.

The following result on irreducible sets in product spaces will be used in the
sequel.

######  Proposition 2.10.

[2, Proposition 8.4.7] Let $\\{X_{i}:i\in I\\}$ be a family of topological
spaces. The irreducible closed sets in $\prod_{i\in I}X_{i}$ are exactly the
sets of the form $\prod_{i\in I}C_{i}$, where each $C_{i}$ is an irreducible
closed set in $X_{i}$ ($i\in I$).

The relations among well-filtered spaces, open well-filtered spaces and sober
spaces are shown below:

soberwell-filteredopen well-filtered core-compact

The following lemma will be used in the sequel.

######  Lemma 2.11.

[6, Lemma 2.4] Let $X$ be a $T_{0}$ space, $\\{U_{i}:i\in I\\}$ be a
$\ll$-filtered family of open sets of $X$, and $F$ be a closed set of $X$. If
$F\cap U_{i}\not=\emptyset$ for all $i\in I$, then there is a minimal closed
set $F_{0}\subseteq F$ such that $F_{0}\cap U_{i}\not=\emptyset$ for all $i\in
I$. In addition, this $F_{0}$ is irreducible.

##  3 A new characterization for open well-filtered spaces

In [7, 9], a characterization for well-filtered spaces by means of KF-sets is
obtained. We now prove a similar characterization for open well-filtered
spaces.

######  Definition 3.1.

A nonempty subset $A$ of a $T_{0}$ space $X$ is called an _open well-filtered
set_ , or _OWF-set_ for short, if there exists a $\ll$-filtered family
$\\{U_{i}:i\in I\\}\subseteq\mathcal{O}(X)$ such that ${\rm cl}(A)$ is a
minimal closed set that intersects all $U_{i}$, $i\in I$.

Denote by $\mathsf{OWF}(X)$ the set of all closed OWF-subsets of $X$.

######  Remark 3.2.

  1. (1)

A subset of a topological space is an OWF-set if and only if its closure is an
OWF-set.

  2. (2)

By Lemma 2.11, every OWF-set is irreducible.

######  Theorem 3.3.

Let $X$ be a $T_{0}$ space. Then the following statements are equivalent:

  1. (1)

$X$ is open well-filtered;

  2. (2)

$\forall A\in\mathsf{OWF}(X)$, there exists a unique $x\in X$ such that
$A={\rm cl}(\\{x\\})$.

######  Proof 3.4.

(1) $\Rightarrow$ (2). Let $A\in\mathsf{OWF}(X)$. Then there exists
$\\{U_{i}:i\in I\\}\subseteq_{flt}\mathcal{O}(X)$ such that $A$ is a minimal
closed set that intersects all $U_{i}$, $i\in I$. Since $X$ is open well-
filtered, it follows that $\bigcap_{i\in I}U_{i}\cap A\neq\emptyset$. Choose
one $x\in\bigcap_{i\in I}U_{i}\cap A$. Then ${\rm cl}(\\{x\\})\subseteq A$,
and it is a closed set that intersects all $U_{i}$, $i\in I$. By the
minimality of $A$, we have that $A={\rm cl}(\\{x\\})$. The uniqueness of $x$
is determined by the $T_{0}$ separation of $X$.

(2) $\Rightarrow$ (1). Let $\\{U_{i}:i\in I\\}\subseteq_{flt}\mathcal{O}(X)$
and $U\in\mathcal{O}(X)$ such that $\bigcap_{i\in I}U_{i}\subseteq U$. We need
to show that $U_{i}\subseteq U$ for some $i\in I$. If, on the contrary, that
$U_{i}\nsubseteq U$ for all $i\in I$, then by Lemma 2.11, there exists a
minimal (irreducible) closed set $A\subseteq X\setminus U$ that intersects all
$U_{i}$, $i\in I$. Thus $A\in\mathsf{OWF}(X)$. By the assumption, there exists
a unique $x\in X$ such that $A={\rm cl}(\\{x\\})$. For each $i\in I$, since
${\rm cl}(\\{x\\})\cap U_{i}=A\cap U_{i}\neq\emptyset$, it follows that $x\in
U_{i}$, which implies that $x\in\bigcap_{i\in I}U_{i}\subseteq U$. Thus $x\in
U\cap A$, which contradicts $A\subseteq X\setminus U$. This contradiction
completes the proof.

######  Example 3.5.

Let $\mathbb{N}^{+}$ be the set of all positive integers, $\mathbb{N}^{+}_{\rm
cof}$ be the space of $\mathbb{N}^{+}$ with the co-finite topology (the open
sets are $\emptyset$ and all the complements of finite subsets of
$\mathbb{N}^{+}$), and let $\mathbb{N}^{+}_{\alpha}$ be the Alexandoff space
of $\mathbb{N}^{+}$ (the open sets are the upper subsets of $\mathbb{N}^{+}$
with the usual order of numbers). We have the following claims:

  1. (c1)

It is trivial to check that each subset of $\mathbb{N}^{+}$ is compact in both
$\mathbb{N}^{+}_{\rm cof}$ and $\mathbb{N}^{+}_{\alpha}$. We then deduce that
$U\ll V$ iff $U\subseteq V$ for any open sets $U,V$ in $\mathbb{N}^{+}_{\rm
cof}$ or $\mathbb{N}^{+}_{\alpha}$.

  2. (c2)

Neither $\mathbb{N}^{+}_{cof}$ nor $\mathbb{N}^{+}_{\alpha}$ is open well-
filtered.

For each $n\in\mathbb{N}^{+}$,
$U_{n}=\mathbb{N}^{+}\setminus\\{1,2,3,\ldots,n\\}$ is a nonempty open set in
both $\mathbb{N}^{+}_{\rm cof}$ and $\mathbb{N}^{+}_{\alpha}$. From (c1), it
follows that the family $\\{U_{n}:n\in\mathbb{N}^{+}\\}$ is $\ll$-filtered,
but $\bigcap_{n\in\mathbb{N}^{+}}U_{n}=\emptyset$. By Proposition 2.4, we
obtain (c2).

The following example shows that a saturated subspace of an open well-filtered
space need not be open well-filtered.

######  Example 3.6.

Let $\mathbb{J}=\mathbb{N}^{+}\times(\mathbb{N}^{+}\cup\\{\omega\\})$ be the
Johnstone’s dcpo [2, 3], which is ordered by
$(m,n)\leq(m^{\prime},n^{\prime})$ iff either $m=m^{\prime}$ and $n\leq
n^{\prime}$, or $n^{\prime}=\omega$ and $n\leq m^{\prime}$ (refer to Figure
1).

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

We have the following conclusions.

  1. (r1)

$\Sigma\mathbb{J}$ is open well-filtered.

Note that $\forall U,V\in\sigma(\mathbb{J})$, $U\ll V$ iff $U=\emptyset$ (see
[2, Exercise 5.2.15]), which implies that each $\ll$-filtered family
$\mathcal{F}$ of $\sigma(\mathbb{J})$ is equal to $\\{\emptyset,U\\}$, where
$U$ is an arbitrary Scott open set in $\mathbb{J}$. This means there exists no
OWF-set in $\Sigma\mathbb{J}$. By Theorem 3.3, we deduce that
$\Sigma\mathbb{J}$ is open well-filtered.

  2. (r2)

The set of maximal points $\mathbb{N}^{+}\times\\{\omega\\}$, as a saturated
subspace of $\Sigma\mathbb{J}$, is homeomorphic to $\mathbb{N}^{+}_{cof}$ of
Example 3.5, and thus is not open well-filtered.

From (r1), we have that $\mathord{\downarrow}x$ (which is exactly the closure
of $\\{x\\}$ in $\Sigma\mathbb{J}$) is not an OWF-set for each $x\in J$. We
then deduce that the closure of singletons need not be OWF-sets. From (r2), it
follows that the saturated subspace of an open well-filtered spaces need not
be open well-filtered.

######  Remark 3.7.

From Example 3.6, we deduce that if each $\ll$-filtered family of open sets in
a space $X$ contains the empty set, then $X$ must be open well-filtered.

The following example shows that neither the closed subspace nor the retract
of an open well-filtered space is open well-filtered in general.

######  Example 3.8.

[6, Example 4.13] Let $P=\mathbb{J}\cup\mathbb{N}^{+}$, where $\mathbb{J}$ is
the Johnstone’s dcpo. For any $x,y\in P$, define $x\leq y$ if one of the
following conditions holds (refer to Figure 2):

  1. (i)

$x,y\in\mathbb{N}^{+}$ and $x\leq y$ in $\mathbb{N}^{+}$ with the usual
ordering;

  2. (ii)

$x,y\in\mathbb{J}$ and $x\leq y$ in $\mathbb{J}$;

  3. (iii)

$x\in\mathbb{N}^{+}$ and $y=(x,\omega)$.

Figure 2: The poset $P$ of Example 3.8

We have the following conclusions.

  1. (1)

$\Sigma P$ is open well-filtered (see [6, Example 4.13] for details).

  2. (2)

$\mathbb{N}^{+}$ is a Scott closed subset of $P$, and as a subspace of $P$, is
homeomorphic to $\mathbb{N}^{+}_{\alpha}$, hence is not open well-filtered (by
Example 3.5 (2)).

  3. (3)

$\mathbb{N}^{+}$, as a subspace of $P$, is a retract of $P$.

Let $e:\mathbb{N}^{+}\longrightarrow P$ be the identity embedding, and define
$r:P\longrightarrow\mathbb{N}^{+}$ as follows: $\forall x\in P$, $\forall
n\in\mathbb{N}^{+}$, $r(x)=n$ iff

$\left\\{\begin{array}[]{lll}x\in\mathord{\downarrow}(1,\omega),&\mbox{ when
}n=1;\\\
x\in\mathord{\downarrow}(n,\omega)\setminus\mathord{\downarrow}(n-1,\omega),&\mbox{
when }n\geq 2.\end{array}\right.$

Since $\mathbb{N}^{+}$ is a closed set in $\Sigma P$, we have that $e$ is
continuous, and note that
$r^{-1}(\mathord{\downarrow}n)=\mathord{\downarrow}(n,\omega)$ is Scott closed
for each $n\in\mathbb{N}^{+}$, which implies that $r$ is a continuous mapping.
It is clear that the composition $r\circ e$ is the identity mapping on
$\mathbb{N}^{+}$. Thus (3) holds.

From above (1)–(3), we deduce that the closed subspace or the retract of an
open well-filtered space need not be open well-filtered.

######  Proposition 3.9.

Let $X$ be a core-compact space. Then every irreducible set in $X$ is an OWF-
set.

######  Proof 3.10.

Suppose that $A$ is an irreducible subset of $X$. Let

$\mathcal{F}=\\{U\in\mathcal{O}(X):A\cap U\neq\emptyset\\}.$

_Claim 1:_ $\mathcal{F}\subseteq_{flt}\mathcal{O}(X)$.

Let $U_{1},U_{2}\in\mathcal{F}$. Then $A\cap U_{1}\neq\emptyset$ and $A\cap
U_{2}\neq\emptyset$, implying that $A\cap U_{1}\cap U_{2}\neq\emptyset$. Take
$x\in A\cap U_{1}\cap U_{2}$. Since $X$ is core-compact, there exists
$U_{3}\in\mathcal{O}(X)$ such that $x\in U_{3}\ll U_{1}\cap U_{2}$. Note that
$x\in U_{3}\cap A\neq\emptyset$, implying that $U_{3}\in\mathcal{F}$. Thus
$\mathcal{F}$ is $\ll$-filtered.

_Claim 2:_ ${\rm cl}(A)$ is a minimal closed set that intersects all members
of $\mathcal{F}$.

Suppose $B$ is a closed set such that $B\subseteq{\rm cl}(A)$ and $B\cap
U\neq\emptyset$ for all $U\in\mathcal{F}$. We need to prove ${\rm
cl}(A)\subseteq B$. Otherwise, ${\rm cl}(A)\nsubseteq B$, which implies
$A\cap(X\setminus B)\neq\emptyset$. Thus $X\setminus B\in\mathcal{F}$, which
contradicts that $B$ intersects all members of $\mathcal{F}$. Therefore, ${\rm
cl}(A)=B$.

All this shows that ${\rm cl}(A)\in\mathsf{OWF}(X)$.

The following result is immediate by using Proposition 3.9.

######  Corollary 3.11.

[4, 9] Every core-compact open well-filtered space is sober.

The following example shows that the continuous image of an OWF-set need not
be an OWF-set.

######  Example 3.12.

Let $X$ be the Alexandroff space of the Johnstone’s dcpo $\mathbb{J}$ (whose
open sets are the upper sets), and let $f:X\longrightarrow\Sigma\mathbb{J}$ be
the identity mapping. Clearly, $f$ is a continuous mapping. Note that for any
$x\in\mathbb{J}$, since $X$ is locally compact (hence core-compact), by
Proposition 3.9, $\mathord{\downarrow}x$ is an OWF-set in $X$, but it is not
an OWF-subset of $\Sigma\mathbb{J}$ by Example 3.6 (r1).

For a $T_{0}$ space $X$ and $Y\in\mathcal{O}(X)$, if $U,V\in\mathcal{O}(Y)$,
then we have that $U,V\in\mathcal{O}(X)$, and that $U\ll V$ in
$(\mathcal{O}(Y),\subseteq)$ if and only if $U\ll V$ in
$(\mathcal{O}(X),\subseteq)$. Using this fact, one can prove the following
result easily.

######  Proposition 3.13.

Every open subspace of an open well-filtered space is also open well-filtered.

######  Definition 3.14.

[2] A $T_{0}$ space $X$ is called _core-coherent_ if for any
$U,V,W\in\mathcal{O}(X)$, $U\ll V$ implies that $U\cap W\ll V\cap W$.

Example 3.6 shows that singletons need not be OWF-sets. As a remedy, we have
the following result.

######  Theorem 3.15.

Let $X$ be a core-coherent space. Then the following statements are
equivalent:

  1. (1)

$X$ is core-compact;

  2. (2)

all irreducible sets in $X$ are OWF-sets;

  3. (3)

the singletons are OWF-sets.

######  Proof 3.16.

By Proposition 3.9, that (1) $\Rightarrow$ (2) is clear, and since singletons
are irreducible, we obtain that (2) $\Rightarrow$ (3). Now we prove (3)
$\Rightarrow$ (1).

Let $x\in X$ and $U$ be an open neighborhood of $x$. Since $\\{x\\}$ is an
OWF-set, there exists a $\ll$-filtered family $\\{U_{i}:i\in
I\\}\subseteq\mathcal{O}(X)$ such that ${\rm cl}(\\{x\\})$ is a minimal closed
set that intersects all $U_{i}$, $i\in I$. Fix an $i_{0}\in I$. It follows
that $x\in U_{i_{0}}$. Then there exists ${i_{1}}\in I$ such that $x\in
U_{i_{1}}\ll U_{i_{0}}$. Since $X$ is core-coherent, it holds that $x\in U\cap
U_{i_{1}}\ll U\cap U_{i_{0}}\subseteq U$. Therefore, $X$ is core-compact.

It is easy to verify the following lemma.

######  Lemma 3.17.

Let $f:X\longrightarrow Y$ be a continuous open mapping between $T_{0}$
spaces, and $U,V\in\mathcal{O}(X)$. If $U\ll V$, then $f(U)\ll f(V)$.

Regarding the product spaces, we are still not able to prove that the product
of two open well-filtered spaces is a open well-filtered space. Here is a
result for some special spaces.

######  Theorem 3.18.

For each $T_{0}$ space $X$, the product $X\times\Sigma\mathbb{J}$ is open
well-filtered.

######  Proof 3.19.

Let $U,V\in\mathcal{O}(X\times\Sigma\mathbb{J})$ such that $U\ll V$. We prove
that $U=\emptyset$. Note that the projection $p_{2}$ is a continuous open
mapping, so $p_{2}(U),p_{2}(V)\in\sigma(\mathbb{J})$ and by Lemma 3.17,
$p_{2}(U)\ll p_{2}(V)$. Thus $p_{2}(U)=\emptyset$, which implies that
$U=\emptyset$. By Remark 3.7, $X\times\Sigma\mathbb{J}$ is an open well-
filtered space.

The above theorem indicates that the open well-filteredness of the product of
spaces does not imply the open well-filteredness of the factor spaces. In the
following, we will show that the open well-filteredness of the product of
finite $T_{0}$ spaces implies that one of the factor spaces is open well-
filtered.

It also is trivial to verify the following lemma.

######  Lemma 3.20.

Let $\\{X_{k}:1\leq k\leq n\\}$ be a finite family of $T_{0}$ spaces, and
$U_{k},V_{k}\in\mathcal{O}(X)$ such that $U_{k}\ll V_{k}$ for $1\leq k\leq n$.
Then $\prod_{1\leq k\leq n}U_{k}\ll\prod_{1\leq k\leq n}V_{k}$.

######  Theorem 3.21.

Let $\\{X_{k}:1\leq k\leq n\\}$ be a finite family of $T_{0}$ spaces, and $X$
be their product space. If $X$ is an open well-filtered space, then there is
$k_{0}$ ($1\leq k_{0}\leq n$) such that $X_{k_{0}}$ is open well-filtered.

######  Proof 3.22.

Suppose on the contrary that every $X_{k}$ is not open well-filtered, $1\leq
k\leq n$. Then there exist a $\ll$-filtered family
$\mathcal{F}_{k}\subseteq\mathcal{O}(X_{k})$ and an open set $O_{k}$ in
$X_{k}$ such that $\bigcap\mathcal{F}_{k}\subseteq O_{k}$, but $U\nsubseteq
O_{k}$ for any $U\in\mathcal{F}_{k}$. Define

$\mathcal{F}=\left\\{\prod_{1\leq k\leq n}U_{k}:\forall
k,U_{k}\in\mathcal{F}_{k}\right\\}.$

By Lemma 3.20, one can deduce that $\mathcal{F}$ is a $\ll$-filtered family of
$\mathcal{O}(X)$ such that $\bigcap\mathcal{F}\subseteq\prod_{1\leq k\leq
n}O_{k}\in\mathcal{O}(X)$. Since $X$ is open well-filtered, there exists
$\prod_{1\leq k\leq n}U_{k}\in\mathcal{F}$ (i.e., $U_{k}\in\mathcal{F}_{k}$
for $1\leq k\leq n$) such that $\prod_{1\leq k\leq
n}U_{k}\subseteq\prod_{1\leq k\leq n}O_{k}$. Note that each $U_{k}$ ($1\leq
k\leq n$) is not empty, which follows that $U_{k}\subseteq O_{k}$ for $1\leq
k\leq n$, a contradiction.

##  4 Upper spaces and open well-filteredness

In this section, we prove that if a space $X$ is open well-filtered then its
upper space (or the Smyth power space) is also open well-filtered. The proof
here makes use of a technique employed in [5].

For any topological space $X$, we use $\mathcal{D}(X)$ to denote the set of
all nonempty compact saturated subsets of $X$. The _upper Vietoris topology_
on $\mathcal{D}(X)$ is the topology that has $\\{\Box U:U\in\mathcal{O}(X)\\}$
as a base, where $\Box U=\\{K\in\mathcal{D}(X):K\subseteq U\\}$. The set
$\mathcal{D}(X)$ equipped with the upper Vietoris topology is called the
_upper space_ or _Smyth power space_ of $X$. Note that
$\\{\diamondsuit\,F:X\setminus F\in\mathcal{O}(X)\\}$ is a base of the co-
upper Vietoris topology, where $\diamondsuit\,F=\\{K\in\mathcal{D}(X):K\cap
F\not=\emptyset\\}$.

######  Remark 4.1.

Let $X$ be a $T_{0}$ space, and $U,U_{1},U_{2}\in\mathcal{O}(X)$.

  1. (1)

$\Box U_{1}\subseteq\Box U_{2}$ if and only if $U_{1}\subseteq U_{2}$.

  2. (2)

For any $x\notin U$, $\diamondsuit{\rm cl}(\\{x\\})\cap\Box U=\emptyset$.

  3. (3)

If $\Box U\subseteq\Box U_{1}\cup\Box U_{2}$, then $\Box U\subseteq\Box U_{1}$
or $\Box U\subseteq\Box U_{2}$ [5].

We now state and prove the main result in this section.

######  Theorem 4.2.

For any open well-filtered space $X$, the upper space $\mathcal{D}(X)$ is open
well-filtered.

######  Proof 4.3.

Assume that $X$ is an open well-filtered space.

Let $\\{\mathcal{U}_{i}:i\in I\\}$ be a $\ll$-filtered family of open sets
$\mathcal{U}_{i}$ in $\mathcal{D}(X)$ and

$\bigcap\\{\mathcal{U}_{i}:i\in I\\}\subseteq\mathcal{U}$

for an open set $\mathcal{U}$ in $\mathcal{D}(X)$.

For each $i\in I$, let $\mathcal{U}_{i}=\bigcup\\{\Box U_{i,t}:t\in T_{i}\\}$,
where $U_{i,t}\in\mathcal{O}(X)$.

Assume that for each $i\in I$, $\mathcal{U}_{i}\not\subseteq\mathcal{U}$. Let
$\mathcal{C}=\mathcal{D}(X)\setminus\mathcal{U}$. Then $\mathcal{C}$ is a
closed set in $\mathcal{D}(X)$ such that
$\mathcal{C}\cap\mathcal{U}_{i}\neq\emptyset$ for all $i\in I$. By Lemma 2.11,
there is a minimal closed set $\mathcal{C}_{0}\subseteq\mathcal{C}$ which also
has a nonempty intersection with every $\mathcal{U}_{i}$.

For each $i\in I$, let

$\hat{\mathcal{U}}_{i}=\bigcup\\{\Box U_{i,t}:\Box
U_{i,t}\cap\mathcal{C}_{0}\not=\emptyset,t\in T_{i}\\}.$

_Fact 1:_ If
$\mathcal{U}_{i_{1}}\ll\mathcal{U}_{i_{2}}\ll\mathcal{U}_{i_{3}}$, then
$\hat{\mathcal{U}}_{i_{1}}\ll\hat{\mathcal{U}}_{i_{3}}$. Therefore,
$\\{\hat{\mathcal{U}}_{i}:i\in I\\}$ is $\ll$-filtered.

We just need to verify the first statement. To see this, let
$\\{\mathcal{V}_{l}:l\in L\\}$ be a directed open cover of
$\hat{\mathcal{U}}_{i_{3}}$. Then
$\\{\mathcal{V}_{l}\cup(\mathcal{D}(X)\setminus\mathcal{C}_{0}):l\in L\\}$ is
a directed open cover of $\mathcal{U}_{i_{3}}$. By the assumption, there is a
$l_{0}\in L$ such that
$\mathcal{U}_{i_{2}}\subseteq\mathcal{V}_{l_{0}}\cup(\mathcal{D}(X)\setminus\mathcal{C}_{0})$.

By $\mathcal{U}_{i_{1}}\ll\mathcal{U}_{i_{2}}$ and the structure of the upper
Vietoris topology, there exist

$\Box W_{1},\Box W_{2},\cdots,\Box W_{n}\mbox{ contained in
}\mathcal{V}_{l_{0}},$ $\Box G_{1},\Box G_{2},\cdots,\Box G_{m}\mbox{
contained in }\mathcal{D}(X)\setminus\mathcal{C}_{0}$

such that

$\mathcal{U}_{i_{1}}\subseteq\bigcup\\{\Box W_{k}:1\leq k\leq
n\\}\cup\bigcup\\{\Box G_{h}:1\leq h\leq m\\}.$

Note that $\hat{\mathcal{U}}_{i_{1}}\subseteq\mathcal{U}_{i_{1}}$. For each
$t\in T_{i_{1}}$ such that $\Box U_{i_{1},t}\cap\mathcal{C}_{0}\neq\emptyset$,
by Remark 4.1, $\Box U_{i_{1},t}\subseteq\Box W_{k}$ for some $1\leq k\leq n$,
or $\Box U_{i_{1},t}\subseteq\Box G_{h}$ for some $1\leq h\leq m$. But $\Box
G_{h}\cap\mathcal{C}_{0}=\emptyset$, hence $\Box U_{i_{1},t}\subseteq\Box
W_{k}\subseteq\mathcal{V}_{l_{0}}$ for some $k$. It then follows that

$\hat{\mathcal{U}}_{i_{1}}\subseteq\mathcal{V}_{l_{0}}.$

Therefore $\hat{\mathcal{U}}_{i_{1}}\ll\hat{\mathcal{U}}_{i_{3}}$ holds.

For each $i\in I$, let

$U_{i}=\bigcup\\{U_{i,t}:\Box U_{i,t}\cap\mathcal{C}_{0}\not=\emptyset\\}.$

_Fact 2:_ If $\hat{\mathcal{U}}_{i_{1}}\ll\hat{\mathcal{U}}_{i_{2}}$, then
$U_{i_{1}}\ll U_{i_{2}}.$ Hence $\\{U_{i}:i\in I\\}$ is a $\ll$-filtered
family of open sets in $X$.

As a matter of fact, it is easy to see that if $\\{W_{j}:j\in
J\\}\subseteq\mathcal{O}(X)$ is a directed open cover of $U_{i_{2}}$, then
$\\{\Box W_{j}:j\in J\\}$ is a directed open cover of
$\hat{\mathcal{U}}_{i_{2}}$, hence there is $j_{0}\in J$ such that
$\hat{\mathcal{U}}_{i_{1}}\subseteq\Box W_{j_{0}}$, thus $U_{i_{1}}\subseteq
W_{j_{0}}$.

Let

$K=\bigcap\\{U_{i}:i\in I\\}.$

By Proposition 2.4, $K$ is a nonempty saturated compact set, that is
$K\in\mathcal{D}(X)$.

_Fact 3:_ $K\not\in\mathcal{U}$.

Indeed, if $K\in\mathcal{U}$, then there is an open set $E$ of $X$ such that
$K\in\Box E\subseteq\mathcal{U}$. Then $K=\bigcap\\{U_{i}:i\in I\\}\subseteq
E$, and since $X$ is open well-filtered, there is $i_{0}$ such that
$U_{i_{0}}\subseteq E$.

Choose one $U_{i_{0},t_{0}}$ such that $\Box
U_{i_{0},t_{0}}\cap\mathcal{C}_{0}\not=\emptyset$. Then
$U_{i_{0},t_{0}}\subseteq U_{i_{0}}\subseteq E$, and it follows that

$\emptyset\neq\Box U_{i_{0},t_{0}}\cap\mathcal{C}_{0}\subseteq\Box
E\cap\mathcal{C}\subseteq\mathcal{U}\cap\mathcal{C}=\emptyset,$

a contradiction.

_Fact 4:_ $K\in\bigcap\\{\hat{\mathcal{U}}_{i}:i\in I\\}$.

As a matter of fact, if the statement is not true, then there is $i_{0}\in I$
such that $K\not\in\hat{\mathcal{U}}_{i_{0}}$. By the definition of
$\hat{\mathcal{U}}_{i_{0}}$. Take any $U_{i_{0},t_{0}}$ with $\Box
U_{i_{0},t_{0}}\cap\mathcal{C}_{0}\not=\emptyset$. Then $K\not\subseteq
U_{i_{0},t_{0}}$. Take an $e\in K\setminus U_{i_{0},t_{0}}$ and let
$\mathcal{F}=\diamondsuit{\rm cl}(\\{e\\})$. Then by Remark 4.1
$\mathcal{F}\cap\Box U_{i_{0},t_{0}}=\emptyset$.

We show that $\mathcal{C}_{0}\cap\mathcal{F}\cap\mathcal{U}_{i}\not=\emptyset$
for all $i\in I$. For this, it is enough to show that for any $i\in I$, there
is a $t_{i}\in T_{i}$ such that

$\Box U_{i,t_{i}}\cap\mathcal{F}\cap\mathcal{C}_{0}\not=\emptyset.$

If not, there exists $i_{1}\in I$ such that

$\Box
U_{i_{1},t}\subseteq(\mathcal{D}(X)\setminus\mathcal{F})\cup(\mathcal{D}(X)\setminus\mathcal{C}_{0})$

for all $t\in T_{i_{1}}$. Choose $i_{2}$ such that
$\hat{\mathcal{U}}_{i_{2}}\ll\hat{\mathcal{U}}_{i_{1}}$. Then there are

$\Box V^{1}_{1},\Box V^{1}_{2},\cdots,\Box V^{1}_{m}\mbox{ contained in
}\mathcal{D}(X)\setminus\mathcal{C}_{0},$

and

$\Box V^{2}_{1},\Box V^{2}_{2},\cdots,\Box V^{2}_{n}\mbox{ contained in
}\mathcal{D}(X)\setminus\mathcal{F}$

such that

$\hat{\mathcal{U}}_{i_{2}}\subseteq\bigcup\\{\Box V^{1}_{k}:1\leq k\leq
m\\}\cup\bigcup\\{\Box V^{2}_{l}:1\leq l\leq n\\}.$

By the definition of $K$, we have that $e\in K\subseteq U_{i_{2}}$, and then
there is $t^{\prime}\in T_{i_{2}}$ such that $e\in U_{i_{2},t^{\prime}}$ where
$\Box U_{i_{2},t^{\prime}}\cap\mathcal{C}_{0}\not=\emptyset$. Thus
$Sat_{X}(\\{e\\})\in\mathcal{F}\cap\Box U_{i_{2},t^{\prime}}\neq\emptyset$.

Now $\Box
U_{i_{2},t^{\prime}}\subseteq\hat{\mathcal{U}}_{i_{2}}\subseteq\bigcup\\{\Box
V^{1}_{k}:1\leq k\leq m\\}\cup\bigcup\\{\Box V^{2}_{l}:1\leq l\leq n\\}$. By
Remark 2.1, we have that

$\Box U_{i_{2},t^{\prime}}\subseteq\Box V^{1}_{k}\mbox{ for some }1\leq k\leq
m,\mbox{ or }\Box U_{i_{2},t^{\prime}}\subseteq\Box V^{2}_{l}\mbox{ for some
}1\leq l\leq n.$

But for any $k$, since $\Box
U_{i_{2},t^{\prime}}\cap\mathcal{C}_{0}\neq\emptyset$ and $\Box
V^{1}_{k}\cap\mathcal{C}_{0}=\emptyset$, it follows that $\Box
U_{i_{2},t^{\prime}}\nsubseteq\Box V^{1}_{k}$, and for any $l$, since $\Box
U_{i_{2},t^{\prime}}\cap\mathcal{F}\neq\emptyset$ and $\Box
V^{2}_{l}\cap\mathcal{F}=\emptyset$, it follows that $\Box
U_{i_{2},t^{\prime}}\nsubseteq\Box V^{2}_{l}$, a contradiction. This shows
that $\mathcal{C}_{0}\cap\mathcal{F}\cap\mathcal{U}_{i}\not=\emptyset$ for all
$i\in I$.

By the minimality of $\mathcal{C}_{0}$, we have that
$\mathcal{C}_{0}\subseteq\mathcal{F}$. Then $\Box
U_{i_{0},t_{0}}\cap\mathcal{C}_{0}\subseteq\Box
U_{i_{0},t_{0}}\cap\mathcal{F}$. But, as we pointed out earlier, $\Box
U_{i_{0},t_{0}}\cap\mathcal{F}=\emptyset$, thus $\Box
U_{i_{0},t_{0}}\cap\mathcal{C}_{0}=\emptyset$. This contradicts that $\Box
U_{i_{0},t_{0}}\cap\mathcal{C}_{0}\not=\emptyset$.

All these together show that $K\in\bigcap\\{\hat{\mathcal{U}}_{i}:i\in I\\}$.

Now, Fact 3 and Fact 4 contradict the assumption

$\bigcap\\{\mathcal{U}_{i\in I}:i\in I\\}\subseteq\mathcal{U}.$

The proof is thus completed.

##  5 Summary

In this paper, we mainly considered the preservation of the open well-
filteredness by some standardly constructed spaces from an open well-filtered
space. The table below summarizes the main results, where “sp.” denotes
“subspaces”.

open sp. | closed sp. | saturated sp. | retract | upper space | product
---|---|---|---|---|---
$\checkmark$ | $\times$ | $\times$ | $\times$ | $\checkmark$ | $?$

Note that in many other cases, the ground spaces usually inherit the
corresponding properties of their upper spaces as they can be embedded into
the upper spaces under the principle filter mappings. At the moment, the
following problem is still open.

######  Problem 5.1.

Is it true that a space is open well-filtered if its upper space is open well-
filtered?

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