HNCO}/N_{\rm SiO}}$, ${N_{\rm HNCO}/N_{\rm HC_{3}N}}$ and ${N_{\rm
HC_{3}N}/N_{\rm SiO}}$ show no obvious trends.

_G001.226 $+$00.095_. The HC3N emission peak has an offset from the continuum
and SiO emission peak by about a beam ($\sim$38′′, Fig. 20). In the dust
temperature map the arc-like low dust temperature region coincides with the
HNCO emission. The SiO 2-1 line shows obvious redshifts of 8.2 km s-1 and 4.1
km s-1 relative to the $v_{\rm LSR}$ of the HNCO 404–303 and HC3N 10–9 lines.
Abundance ratios of ${N_{\rm HC_{3}N}/N_{\rm SiO}}$ show an increasing and
${N_{\rm HNCO}/N_{\rm SiO}}$ and ${N_{\rm HNCO}/N_{\rm HC_{3}N}}$ a decreasing
trend towards the yellow cross position, respectively.

_G001.344 $+$00.258_. A similarly centrally condensed structure is seen in
HNCO, SiO and HC3N emission in the top panels of Fig. 21. All molecular line
emissions peak towards the low temperature maximum of the 870 $\mu$m dust
continuum emission. The spectra of HNCO 404–303, SiO 2–1 and HC3N 10–9 show a
single velocity component. The trend of ${N_{\rm HC_{3}N}/N_{\rm SiO}}$ is
opposite to that of the dust emission temperature.

_G001.381 $+$00.201_. From the lower panels of Fig. 21, it can be seen that
both the integrated intensity maps of HNCO and HC3N show two compact cores. In
addition, the upper compact cores of SiO, HNCO and HC3N do not coincide with
each other. Moreover, compared to the south-eastern core of HNCO and HC3N, SiO
emission shows more extended and complex structure. Abundance ratios and dust
temperature appear not to be clearly correlated.

_G001.510 $+$00.155_. In the top panels of Fig. 22, the SiO, HNCO and HC3N
emissions clearly show three, two, and one compact core(s), respectively. The
HC3N core is elongated from the south-east to the north-west, and coincides
well with the low dust temperature region. The SiO 2–1, HNCO 404–303 and HC3N
10–9 lines can be fitted by a single Gaussian component. The abundance ratio
curve of ${N_{\rm HC_{3}N}/N_{\rm SiO}}$ shows an obvious plateau near the
yellow cross position along the Galactic latitude direction.

_G001.610 $-$00.172_ and _G001.655 $-$00.062_. From Figs. 22 and 23, it can be
seen that both sources present condensed structures in the HNCO 404–303 and
HC3N 10–9 emissions in the low dust temperature region. The peaks of HNCO
404–303 emission show offsets of $\sim$40′′ and $\sim$55′′ from the strongest
870 $\mu$m dust emission in G001.610$-$00.172 and G001.655$-$00.062,
respectively. The SiO 2–1 emissions for these two sources all show complex
structures. In addition, the SiO 2–1 line of G001.655$-$00.062 exhibits a
double peak, while HNCO and HC3N show a single velocity component at the dip
of SiO 2–1. The abundance ratio curve of ${N_{\rm HNCO}/N_{\rm SiO}}$ and
${N_{\rm HC_{3}N}/N_{\rm SiO}}$ show a reverse trend to that of the dust
emission temperature.

_G001.694 $-$00.385_. A north-south elongated and compact structure is shown
in the HNCO 404–303 and HC3N 10–9 emissions, which contains three cores (Fig.
23). These coincide with the region of low temperature 870 $\mu$m dust
emission. The SiO emission shows an extended complex structure also associated
with the region of higher dust temperature in the south-east. The spectrum of
SiO 2-1 is noisy. Near the yellow cross position, ${N_{\rm HC_{3}N}/N_{\rm
SiO}}$ reaches maxima, while ${N_{\rm HNCO}/N_{\rm SiO}}$ and ${N_{\rm
HNCO}/N_{\rm HC_{3}N}}$ and the dust temperature are low.

_G001.699 $-$00.366_. Two compact cores can be seen in the integrated
intensity maps of HNCO 404–303 and HC3N 10–9 in the top panels of Fig. 24.
This is consistent with the distributions of low temperature dust emission.
There are also two cores in the SiO 2–1 line map, but here the emission peaks
are offset from the dust continuum emission peaks ($\sim$35′′). Both of the
HNCO and HC3N lines show single Gaussian velocity component, but SiO shows two
velocity components on either side of the systemic velocity of the source.
${N_{\rm HNCO}/N_{\rm HC_{3}N}}$ follows the trend of the dust temperature.

_G001.734 $-$00.410_ and _G001.883 $-$00.062_. In Figs. 24 and Fig. 25, our
three main molecular tracers show similar morphologies and a centrally
condensed structure. Moreover, obvious offsets of $\sim$15′′ and $\sim$30′′
between the molecular line emission peaks and the lower temperature 870 $\mu$m
dust continuum emission peaks are found in G001.734$-$00.410 and
G001.883$-$00.062, respectively. All spectra show one Gaussian component. Near
the yellow cross position, ${N_{\rm HC_{3}N}/N_{\rm SiO}}$ reaches maxima,
while ${N_{\rm HNCO}/N_{\rm SiO}}$ and ${N_{\rm HNCO}/N_{\rm HC_{3}N}}$ and
the dust temperature are low.

_G003.240 $+$00.635_. Extended emission can be seen in the integrated
intensity maps of the SiO and HC3N lines in the lower panels of Fig. 25. In
addition, HNCO has the most compact emission morphology, and coincides with
the 870 $\mu$m emission peak, which shows the lowest dust temperature. The
HNCO and SiO lines are Gaussian towards the yellow cross position. All three
plotted abundance ratios follow the trend determined by the dust temperature.

_G003.338 $+$00.419_. The top panels of Fig. 26 show a compact core, which is
seen in the 870 $\mu$m continuum and the HNCO and HC3N emission. SiO shows an
overall similar morphology as HC3N but its distribution on smaller scales
appear to be more complex. The SiO 2–1 line shows a skewed profile with a
blue-shifted peak that is weaker than the red-shifted one. Near the yellow
cross position, ${N_{\rm HC_{3}N}/N_{\rm SiO}}$ reaches maxima, while ${N_{\rm
HNCO}/N_{\rm SiO}}$ and ${N_{\rm HNCO}/N_{\rm HC_{3}N}}$ and the dust
temperature are low.

_G008.671 $-$00.357_. Fig. 27 shows a bubble, CN129, nearby, in the north-west
of the presented maps of integrated line emission. This corresponds to a hot
dust core, which is close to a class II CH3OH maser (red diamond shown in the
lower left panel of Fig. 27) (Caswell et al., 1993). The H2O maser (red
asterisk shown in the same panel) lies in between the two dust emission cores,
which indicates a possible interaction. The three molecular lines all show a
morphology extending from north-west to south-east. Two peaks of HC3N emission
are seen towards the dust continuum emission. The trend of ${N_{\rm
HNCO}/N_{\rm SiO}}$ is opposite to that of the dust temperature along the
Galactic latitude direction. Inflow and outflow activities are simultaneously
detected in this source from analysing molecular line profiles of HCO+ 1–0
(see Sect. 5.4 and Fig. 11 for details).

_G010.473 $+$00.028_. From the lower panels of Fig. 27, it can be seen that
the molecular line emissions exhibit centrally condensed structure. SiO and
HNCO emissions peak towards the faint dust continuum emission core, which
contains OH (Caswell, 1998) and class II CH3OH masers (Pestalozzi et al.,
2005; Caswell et al., 1993), while HC3N emission peaks towards the maximum of
hot dust continuum emission. There is a bubble, CN151, close to the peak of
the HC3N emission, and an OH maser between them. Because of the low signal-to-
noise ratio of the SiO 2-1 data, the integrated intensity is calculated in the
wide velocity range between 55.54 to 84.44 km s-1. ${N_{\rm HNCO}/N_{\rm
HC_{3}N}}$ increases with increasing Galactic longitude, while $N_{\rm
HC_{3}N}$/$N_{\rm SiO}$ follows an opposite trend.

_G322.159 $+$00.635_. It is identified as an IRDC of SDC 322.149+0.639 by
Peretto & Fuller (2009). As Fig. 28 shows, the map consists of two hot
molecular cores embedded in bubble MWP1G322154+006351 identified by Simpson et
al. (2012). The north-western molecular core contains OH (Caswell & Haynes,
1987) and CH3OH (Caswell et al., 1993) masers, while the south-eastern
molecular core is close to another OH maser, which lies in between two dust
cores. All three molecular lines show a single velocity component. Abundance
ratios of ${N_{\rm HNCO}/N_{\rm SiO}}$ and ${N_{\rm HNCO}/N_{\rm HC_{3}N}}$
show similarly varying tendencies.

_G326.653 $+$00.618_. In the lower panels of Fig. 28, there is an obvious dust
temperature gradient decreasing from the south-east to the north-west.
Furthermore, there is a bubble, S79, to the south-east. The peaks of molecular
line emission have a significant offset of $\sim$30′′ from the peak of the
south-eastern hot dust clump, which might be caused by the shock between the
bubble and the molecular clouds. The SiO spectrum from the maximum of dust
continuum emission presents an additional velocity component at $\sim-$30 km
s-1. All these lines present two cores located on two sides of the brighest
dust peak also hosting an H2O maser (Batchelor et al., 1980). Abundance ratios
and dust temperature appear not to be clearly correlated.

_G327.293 $-$00.579_. G327.293$-$00.579 is identified as an IRDC, named SDC
327.306$-$0.566 by Peretto & Fuller (2009). As Fig. 29 shows, the spectral
line maps reveal an isolated core. The peak molecular line emissions are
slightly offset by $\sim$20′′ from the maximum of the dust continuum emission,
which shows a 12-GHz CH3OH maser (Caswell et al., 1995) close to the bubble,
MWP1G327281$-$005629\. Because of many saturated pixels in the dust
temperature map, we just show integrated intensity ratios of ${I_{\rm
HNCO}/I_{\rm HC_{3}N}}$, ${I_{\rm HNCO}/I_{\rm SiO}}$ and ${I_{\rm
HC_{3}N}/I_{\rm SiO}}$ and find that they exhibit similar trends.

_G345.004 $-$00.224_. G345.004$-$00.224 is identified as an IRDC, named SDC
345.000$-$0.232 by Peretto & Fuller (2009). As shown in the lower panels of
Fig. 29, molecular line emissions show a similar, centrally condensed
structure. The separation between the bubble centroid and dust emission peak
is $\sim$160″, corresponding to a projected distance of $\sim$1 pc at a
kinematic distance of 1.4 kpc (Urquhart et al., 2018). SiO and HC3N emissions
peak toward the hot dust clump which contains H2O (Caswell et al., 1983),
12-GHz CH3OH (Caswell et al., 1995) and OH (Caswell & Haynes, 1983) masers,
while the HNCO emission peak has an offset of $\sim$20′′ from the maximum of
the dust continuum emission. Moreover, the SiO line shows non-Gaussian line-
wing emission, which indicates outflow activities. Abundance ratios of
${N_{\rm HNCO}/N_{\rm SiO}}$ and ${N_{\rm HNCO}/N_{\rm HC_{3}N}}$ increase
away from the central hot clump. Inflow and outflow activities are
simultaneously detected in this source from analysing the molecular line
profile of HCO+ 1–0 and SiO 2-1, respectively (see Sect. 5.4 and Fig. 11 for
details).

_G350.101 $+$00.083_. G350.101$+$00.083 is identified as an IRDC, named SDC
350.117+0.096 by Peretto & Fuller (2009). In the top panels of Fig. 30, the
SiO, HNCO and HC3N molecular emissions show similar morphologies, with
extended structure associated well with the hot 870 $\mu$m, OH maser (Caswell
& Haynes, 1983) and water vapor maser emission (Caswell et al., 1983). There
are three bubbles, MWP1G350105$+$000838, MWP1G350107$+$000852, which is a
smaller bubble inside the area of MWP1G350105$+$000838, and CS112 (Simpson et
al., 2012) located south-west and south of the molecular emission peak. The
SiO 2-1 spectrum shows a low signal-to-noise ratio. The abundance ratio curve
of ${N_{\rm HNCO}/N_{\rm SiO}}$ and ${N_{\rm HNCO}/N_{\rm HC_{3}N}}$ show a
reverse trend to that of the dust emission temperature.

_G351.443 $+$00.659_. In the lower panels of Fig. 30, the integrated intensity
maps of the SiO 2–1, HNCO 404–303 and HC3N 10–9 lines all show a similar,
centrally condensed structure. The peak molecular line emissions are obviously
offset from the maximum of dust continuum emission by $\sim$15′′, $\sim$30′′
and $\sim$15′′ for SiO, HNCO and HC3N, respectively, which might be caused by
the shocks from bubbles MWP1G351426$+$006596 and MWP1G351418$+$006431\. The
SiO 2–1 line shows broad wings. Abundance ratios of ${N_{\rm HNCO}/N_{\rm
HC_{3}N}}$ and ${N_{\rm HNCO}/N_{\rm SiO}}$ show an opposite trend with
respect to the dust temperature along Galactic latitude. Inflow and outflow
activities are simultaneously detected in this source from analysing the
molecular line profiles of HCO+ 1–0 (see Sect. 5.4 and Fig. 11 for details).

_G351.582 $-$00.352_. In the top panels of Fig. 31, the three molecular
emissions show similar morphologies with a single emission peak towards the
hot dust clump, which contains H2O (Caswell et al., 1983), class II CH3OH
(Caswell et al., 2010) and OH (Caswell & Haynes, 1983) masers. A bubble,
MWP1G351585$-$003560, identified by Simpson et al. (2012) is close to the
maximum of dust continuum emission. The velocity range of the SiO line is the
largest. ${I_{\rm HC_{3}N}/I_{\rm SiO}}$ has an obvious maximum towards the
north-east of the molecular emission peak.

_G351.775 $-$00.537_. In the lower panels of Fig. 31, the three molecular
lines show a similar morphology with emission peaks near the bubble
MWP1G351780$-$005500S. The peak molecular line emission of HC3N is offset by
$\sim$30′′ from the maximum of hot continuum emission. The SiO 2–1 line
profile shows a particularly wide velocity range. The trend of the ${I_{\rm
HNCO}/I_{\rm HC_{3}N}}$ ratio is opposite to that of ${I_{\rm HC_{3}N}/I_{\rm
SiO}}$. Inflow and outflow activities are simultaneously detected in this
source from analysing molecular line profiles of HCO+ 1–0 (see Sect. 5.4 and
Fig. 11 for details).

_G329.030 $-$00.202_. G329.030$-$00.202 is identified as an IRDC, named SDC
329.028$-$0.202 by Peretto & Fuller (2009). As seen in Fig. 32, the
morphologies of our three molecular lines are consistent with that of the 870
$\mu$m hot dust emission. The source contains 12-GHz CH3OH (Caswell et al.,
1995) and H2O (Scalise et al., 1989) masers. The SiO 2–1 line may show a red-
skewed profile with the red-shifted peak being stronger than the blue-shifted
peak. HNCO and and HC3N lines show a single velocity component peaking near
the dip of the SiO 2–1 line. In addition, we note that the SiO line profile
shows wing emission. For abundance ratios of ${N_{\rm HNCO}/N_{\rm SiO}}$ and
${N_{\rm HNCO}/N_{\rm HC_{3}N}}$, there is an obvious decrease towards the hot
dense core, while ${N_{\rm HC_{3}N}/N_{\rm SiO}}$ does the opposite. Moreover,
inflow and outflow activities are simultaneously detected in this source from
analysing molecular line profiles of HCO+ 1–0 (see Sect. 5.4 and Fig. 12 for
details).

_G331.709 $+$00.602 & G331.708$+$00.583_. From the lower panels of Fig. 32, it
can be seen that the three molecular line emissions show similar morphologies,
and are consistent with two hot dust clumps, G331.709+00.602 and
G331.708+00.583, which are indicated by a pink and a yellow cross,
respectively. ${N_{\rm HNCO}/N_{\rm SiO}}$ and ${N_{\rm HNCO}/N_{\rm
HC_{3}N}}$ have a tendency to decrease with dust temperature towards the inner
hot and dense region. Inflow and outflow activities are simultaneously
detected in these two sources from analysing molecular line profiles of HCO+
1–0 (see Sect. 5.4 and Fig. 12 for details).

_G335.586 $-$00.289_. G335.586$-$00.289 is identified as an IRDC, named SDC
335.579$-$0.292 by Peretto & Fuller (2009). As seen in the top panels of Fig.
33, the emissions of the HNCO, SiO and HC3N lines display similar morphologies
with a single core peaking at the hot 870 $\mu$m dust emission clump, which
contains CH3OH (Caswell et al., 1995) and OH (Caswell, 1998) masers. The SiO
line profile shows a peak and a blue shoulder, but the HNCO and HC3N profiles
exhibit single Gaussian components. Similar to sources G331.709+00.602 &
G331.708+00.583, ${N_{\rm HNCO}/N_{\rm SiO}}$ and ${N_{\rm HNCO}/N_{\rm
HC_{3}N}}$ show a decreasing trend towards the warmer inner region. Inflow and
outflow activities are simultaneously detected in this source from analysing
molecular line profiles of HCO+ 1–0 (see Sect. 5.4 and Fig. 12 for details).

_G348.754 $-$00.941_. G348.754$-$00.941 is identified as an IRDC, named SDC
348.765$-$0.924 by Peretto & Fuller (2009). The peaks of the HNCO, SiO and
HC3N emission, showing a cometary distribution, are slightly offset from the
maximum of low temperature filamentary dust continuum emission (Fig. 33). SiO
emission is relatively weak in G348.754$-$00.941. Moreover, the trends of
${N_{\rm HNCO}/N_{\rm SiO}}$ and ${N_{\rm HNCO}/N_{\rm HC_{3}N}}$ are opposite
to that of the dust temperature.

_G351.157 $+$00.701_. In Fig. 34, a north-west – south-east elongated
structure is shown in the three molecular emissions, which contain one core,
with emission peaks obviously offset from the maximum of dust continuum
emission by $\sim$30′′, $\sim$45′′ and $\sim$25′′ for SiO, HNCO and HC3N,
respectively. SiO and HC3N emission peaks towards H2O (Caswell et al., 1983)
and OH (Caswell, 1998) masers where the dust temperature is high. ${I_{\rm
HNCO}/I_{\rm SiO}}$ and ${I_{\rm HC_{3}N}/I_{\rm SiO}}$ show a decreasing
trend towards the yellow cross site.

Figure 13: Contour maps of SiO 2–1, HNCO 404–303 and HC3N 10–9 integrated
intensity superimposed on its 870 $\mu$m continuum emission map in the upper
panels of G359.977$-$00.072 (contours start from 3$\sigma$ and have steps of
3$\sigma$, where 3$\sigma$ is 2.58, 2.10 and 2.22 K km s-1, respectively) and
G359.895$-$00.069 (starting with steps of 6$\sigma$, where 6$\sigma$ is 3.54,
3.30 and 2.94 K km s-1, respectively). Lower panels of G359.977$-$00.072 and
G359.895$-$00.069 left to right: contour maps of SiO 2-1 (white solid lines)
and HNCO 404–303 (red solid lines) integrated intensity superimposed on its
dust temperature map, the extracted beam averaged spectra of SiO 2–1, HNCO
404–303 and HC3N 10–9 from the position marked by a yellow cross, and
normalized variations in ${\rm N_{HNCO}/N_{SiO}}$ (black line), ${\rm
N_{HNCO}/N_{HC_{3}N}}$ (green line), ${\rm N_{HC_{3}N}/N_{SiO}}$ (red line)
and dust temperature (blue dashed line) along the Galactic longitude and
Galactic latitude directions passing through the yellow cross (vertical dashed
lines). The two Gaussian components and total fits to the spectra are shown
with red and green lines, respectively Figure 14: Same as Fig. 13, but for
G359.868$-$00.085 (contours start from 3$\sigma$ with steps of 6$\sigma$,
where 6$\sigma$ is 3.66, 2.94 and 1.92 K km s-1, respectively) and
G359.565$-$00.161 (contours start from 3$\sigma$ with steps of 2$\sigma$,
where 2$\sigma$ is 0.76, 0.96 and 0.80 K km s-1, respectively). Figure 15:
Same as Fig. 13, but for G359.453$-$00.112 (contours start from 3$\sigma$ with
steps of 3$\sigma$, where 3$\sigma$ is 1.23, 1.35 and 1.38 K km s-1,
respectively) and G359.445$-$00.054 (contours start from 3$\sigma$ with steps
of 2$\sigma$, where 2$\sigma$ is 0.84, 1.00 and 0.80 K km s-1, respectively).
Figure 16: Same as Fig. 13, but for G000.067$-$00.077 (contours start from
3$\sigma$ with steps of 3$\sigma$, where 3$\sigma$ is 3.00, 2.85 and 1.44 K km
s-1, respectively) and G000.104$-$00.080 (contours start from 3$\sigma$ with
steps of 5$\sigma$, where 5$\sigma$ is 3.35, 3.50 and 3.15 K km s-1,
respectively). Figure 17: Same as Fig. 13, but for G000.106$-$00.001 (contours
start from 3$\sigma$ with steps of 4$\sigma$, where 4$\sigma$ is 2.08, 2.12
and 1.38 K km s-1, respectively) and G000.110+00.148 (contours start from
3$\sigma$ with steps of 2$\sigma$, where 2$\sigma$ is 1.12, 1.04 and 0.74 K km
s-1, respectively). Figure 18: Same as Fig. 13, but for G000.314$-$00.100
(contours start from 3$\sigma$ with steps of 2$\sigma$, where 2$\sigma$ is
1.84, 1.74 and 1.60 K km s-1, respectively) and G000.497+00.021 (contours
start from 3$\sigma$ with steps of 3$\sigma$, where 3$\sigma$ is 1.80, 1.89
and 1.18 K km s-1, respectively). Figure 19: Same as Fig. 13, but for
G000.645+00.027 (contours start from 3$\sigma$ with steps of 3$\sigma$, where
3$\sigma$ is 2.37, 2.16 and 2.34 K km s-1, respectively) and G000.892+00.143
(contours start from 3$\sigma$ with steps of 2$\sigma$, where 2$\sigma$ is
1.24, 1.24 and 0.96 K km s-1, respectively). Figure 20: Same as Fig. 13, but
for G000.908+00.116 (contours start from 3$\sigma$ with steps of 2$\sigma$,
where 2$\sigma$ is 2.43, 1.58 and 1.44 K km s-1, respectively) and
G001.226+00.059 (contours start from 3$\sigma$ with steps of 2$\sigma$, where
2$\sigma$ is 1.52, 1.46 and 0.96 K km s-1, respectively). Figure 21: Same as
Fig. 13, but for G001.344+00.258 (contours start from 3$\sigma$ with steps of
3$\sigma$, where 3$\sigma$ is 2.19, 2.10 and 1.77 K km s-1, respectively) and
G001.381+00.201 (contours start from 3$\sigma$ with steps of 2$\sigma$,
$\sigma$ and $\sigma$ for SiO 2–1, HNCO 404–303 and HC3N 10–9, respectively.
The corresponding values are 1.02, 0.58 and 0.45 K km s-1). Figure 22: Same as
Fig. 13, but for G001.510+00.155 (contours start from 3$\sigma$ with steps of
3$\sigma$, where 3$\sigma$ is 1.68, 1.56 and 1.11 K km s-1, respectively) and
G001.610$-$00.172 (contours start from 3$\sigma$ with steps of 3$\sigma$,
where 3$\sigma$ is 1.11, 1.11 and 0.78 K km s-1, respectively). Figure 23:
Same as Fig. 13, but for G001.655$-$00.062 (contours start from 3$\sigma$ with
steps of 3$\sigma$, where 3$\sigma$ is 1.26, 1.38 and 1.23 K km s-1,
respectively) and G001.694$-$00.385 (contours start from 3$\sigma$ with steps
of 3$\sigma$, where 3$\sigma$ is 1.86, 2.01 and 1.71 K km s-1, respectively).
Figure 24: Same as Fig. 13, but for G001.699$-$00.366 (contours start from
3$\sigma$ with steps of 3$\sigma$, where 3$\sigma$ is 1.74, 1.50 and 1.41 K km
s-1, respectively) and G001.734$-$00.410 (contours start from 3$\sigma$ with
steps of 3$\sigma$, where 3$\sigma$ is 1.53, 1.35 and 1.08 K km s-1,
respectively). Figure 25: Same as Fig. 13, but for G001.883$-$00.062 (contours
start from 3$\sigma$ with steps of 2$\sigma$, where 2$\sigma$ is 0.72, 0.78
and 0.74 K km s-1, respectively) and G003.240+00.635 (contours start from
3$\sigma$ with steps of 2$\sigma$, where 2$\sigma$ is 1.10, 0.98 and 0.82 K km
s-1, respectively). Figure 26: Same as Fig. 13, but for G003.338+00.419
(contours start from 3$\sigma$ with steps of 2$\sigma$, where 2$\sigma$ is
1.40, 1.12 and 1.04 K km s-1, respectively) and G359.445$-$00.054 (contours
start from 3$\sigma$ with steps of 2$\sigma$, where 2$\sigma$ is 0.84, 1.00
and 0.80 K km s-1, respectively). Figure 27: Same as Fig. 13, but for and
G008.671$-$00.357 (contours start from 3$\sigma$ with steps of $\sigma$,
$\sigma$ and 3$\sigma$ for SiO 2–1, HNCO 404–303 and HC3N 10–9, respectively.
The $\sigma$ values are 0.39, 0.39 and 0.25 K km s-1) and G010.473+00.028
(contours start from 3$\sigma$ with steps of $\sigma$, $\sigma$ and 3$\sigma$
for SiO 2–1, HNCO 404–303 and HC3N 10–9, respectively. The $\sigma$ values are
0.30, 0.31 and 0.34 K km s-1). Figure 28: Same as Fig. 13, but for and
G322.159+00.635 (contours start from 3$\sigma$ with steps of 3$\sigma$,
$\sigma$ and 3$\sigma$ for SiO 2–1, HNCO 404–303 and HC3N 10–9, respectively.
The $\sigma$ values are 0.29, 0.19 and 0.24 K km s-1) and G326.653+00.618
(contours start from 3$\sigma$ with steps of 3$\sigma$, $\sigma$ and 3$\sigma$
for SiO 2–1, HNCO 404–303 and HC3N 10–9, respectively. The $\sigma$ values are
0.26, 0.21 and 0.22 K km s-1). Figure 29: Same as Fig. 13, but for and
G327.293$-$00.579 (contours start from 3$\sigma$ with steps of 3$\sigma$,
$\sigma$ and 3$\sigma$ for SiO 2–1, HNCO 404–303 and HC3N 10–9, respectively.
The $\sigma$ values are 0.24, 0.22 and 0.29 K km s-1) and G345.004$-$00.224
(contours start from 3$\sigma$ with steps of 2$\sigma$, $\sigma$ and 3$\sigma$
for SiO 2–1, HNCO 404–303 and HC3N 10–9, respectively. The $\sigma$ values are
0.43, 0.26 and 0.26 K km s-1). Figure 30: Same as Fig. 13, but for and
G350.101+00.083 (contours start from 3$\sigma$ with steps of 2$\sigma$,
2$\sigma$ and 3$\sigma$ for SiO 2–1, HNCO 404–303 and HC3N 10–9, respectively.
The $\sigma$ values are 0.23, 0.20 and 0.21 K km s-1) and G351.443+00.659
(contours start from 3$\sigma$ with steps of 6$\sigma$, 3$\sigma$ and
5$\sigma$ for SiO 2–1, HNCO 404–303 and HC3N 10–9, respectively. The $\sigma$
values are 0.23, 0.23 and 0.24 K km s-1). Figure 31: Same as Fig. 13, but for
and G351.582$-$00.352 (contours start from 3$\sigma$ with steps of$\sigma$,
3$\sigma$ and 3$\sigma$ for SiO 2–1, HNCO 404–303 and HC3N 10–9, respectively.
The $\sigma$ values are 0.27, 0.24 and 0.27 K km s-1) and G351.775$-$00.537
(contours start from 3$\sigma$ with steps of 3$\sigma$, $\sigma$ and 3$\sigma$
for SiO 2–1, HNCO 404–303 and HC3N 10–9, respectively. The $\sigma$ values are
0.78, 0.35 and 0.38 K km s-1). Figure 32: Same as Fig. 13, but for and
G329.030$-$00.202 (contours start from 3$\sigma$ with steps of 3$\sigma$,
2$\sigma$ and 3$\sigma$ for SiO 2–1, HNCO 404–303 and HC3N 10–9, respectively.
The $\sigma$ values are 0.28, 0.21 and 0.28 K km s-1) and G331.709+00.602 &
G331.708+00.583 (contours start from 3$\sigma$ with steps of $\sigma$,
$\sigma$ and 3$\sigma$ for SiO 2–1, HNCO 404–303 and HC3N 10–9, respectively.
The $\sigma$ values are 0.32, 0.17 and 0.20 K km s-1). Figure 33: Same as Fig.
13, but for and G335.586$-$00.289 (contours start from 3$\sigma$ with steps of
3$\sigma$, $\sigma$ and 3$\sigma$ for SiO 2–1, HNCO 404–303 and HC3N 10–9,
respectively. The $\sigma$ values are 0.25, 0.17 and 0.16 K km s-1) and
G348.754$-$00.941 (contours start from 3$\sigma$ with steps of $\sigma$,
$\sigma$ and 3$\sigma$ for SiO 2–1, HNCO 404–303 and HC3N 10–9, respectively.
The $\sigma$ values are 0.29, 0.26 and 0.22 K km s-1). Figure 34: Same as Fig.
13, but for G351.157+00.701 (contours start from 3$\sigma$ with steps of
3$\sigma$, $\sigma$ and 4$\sigma$ for SiO 2–1, HNCO 404–303 and HC3N 10–9,
respectively. The $\sigma$ values are 0.24, 0.26 and 0.26 K km s-1).


# Chain Conditions and the Axiom of Choice

Ilijas Farah  Department of Mathematics and Statistics
York University
Toronto, Ontario M3J 1P3
CANADA<EMAIL_ADDRESS>and  Jeffrey Marshall-Milne  Department of Mathematics
and Statistics
McMaster University
Hamilton, Ontario L8S 4L8
CANADA<EMAIL_ADDRESS>
(Date: August 2023)

###### Abstract.

Within the framework of Zermelo-Fraenkel set theory without the Axiom of
Choice, we establish equivalents to the assertion "the union of a countable
collection of finite sets is countable" in the context of metric spaces,
probability theory, and Hilbert spaces. The categorical principle underlying
these equivalences is identified.

###### Key words and phrases:

Axiom of Choice, Chain conditions, Metric spaces, Hilbert spaces

Partially supported by NSERC

##  1\. Introduction

Adding to the large body of literature on choice principles and their
equivalents (see [A2] and [A4]), we work in Zermelo-Fraenkel set theory
without the Axiom of Choice (abbreviated ZF). Modern treatments of real
analysis make heavy use of the Axiom of Choice (AC). As a result, we
frequently encounter theorems which may themselves be viewed as choice
principles. A typical example is the equivalence of the Hahn-Banach Extension
Theorem and the Boolean Prime Ideal Theorem. The program of this paper is to
establish more such theorem/choice-principle pairs, typically for fairly weak
choice principles.

As we will see, these arise very naturally when studying chain conditions. A
topological space is said to satisfy the countable chain condition (or it is
simply called ccc) if there does not exist an uncountable collection of
pairwise disjoint open sets. Given a cardinal $\kappa$, there is also the
concept of a $\kappa$-cc topological space, where every collection of pairwise
disjoint open sets has cardinality $<\kappa$. A Boolean algebra is said to be
ccc if the associated Stone space is ccc. A standard ZF+AC argument shows that
a Boolean algebra which admits a probability measure satisfies a rather strong
form of the countable chain condition, called the $\sigma$-bounded chain
condition.

The utility of the countable chain condition (see e.g., [A5]) dates back at
least to P.J. Cohen’s proof of the independence of the Continuum Hypothesis
from ZF+AC. It is commonly used to prove that a forcing notion does not
collapse cardinals or change cofinalities when one passes from a model of
ZF+AC to its forcing extension. When AC is at our disposal, the countable
chain condition admits many useful equivalents in different contexts. For
example, the countable chain condition, separability, and second countability
are all equivalent conditions for metric spaces. The situation becomes more
complicated when we omit the Axiom of Choice (see [A6]).

Serving as a jumping-off point for this project, D.H. Fremlin’s Exercise 561Yc
[A3] shows that these equivalences need not hold when AC fails. In part iii of
this exercise, Fremlin notes that the existence of so called "Russell socks"
implies the existence of a complete and totally bounded metric space which is
non-compact and does not satisfy the countable chain condition (and is
therefore neither separable nor second-countable). This observation is made
precise in 2.4, and is generalized in 2.2. In part iv of the same exercise,
Fremlin notes that the existence of Russell socks further implies the
existence of a probability algebra which is not ccc. 4.1 makes this
implication precise, whereas 5.1 establishes a similar observation in the
context of Hilbert spaces. The categorical principle unifying 2.4, 4.1, and
5.1 is given by 3.2.

This paper is inspired by [i1], [i3], and [i2]. Of note is the closely related
work found in [c3], [c4], [c2], and [c1]. [d1] and [A2] were indispensable
resources in the writing of this paper.

### Acknowledgment

We would like to thank Asaf Karagila for valuable comments on an early version
of this paper, as well as his words of encouragement. This paper would not
have found its current form without his support.

##  2\. Metric Spaces

Let $\kappa$ be a cardinal number (not necessarily well-ordered or infinite).

######  Definition 2.1.

A topological space is said to satisfy the $\kappa^{\leq}$-chain condition if
every collection of pairwise disjoint open sets has cardinality $\leq\kappa$.

Note that the case $\kappa=\aleph_{0}$ corresponds to the countable chain
condition. In presence of the Axiom of Choice, the $\kappa^{\leq}$-chain
condition reduces to the $\kappa^{+}$-chain condition, where $\kappa^{+}$ is
the least cardinal greater than $\kappa$. However, in section 3 we will
consider the $\kappa^{\leq}$ chain condition for cardinals which are not
necessarily well-ordered $\kappa$. Naturally, a metric space is called
$\kappa$-cc (or $\kappa^{\leq}$-cc) if it is $\kappa$-cc (respectively,
$\kappa^{\leq}$-cc) as a topological space. Assertion ii of 2.2 is a standard
theorem of ZF+AC. 2.2 makes its relationship to the Axiom of Choice precise.

######  Theorem 2.2.

Let $\kappa$ be an infinite well-ordered cardinal. The following are
equivalent.

  * i

If $X=(X_{1},X_{2},...)$ is a sequence of sets such that $|X_{n}|<\kappa$ for
all $n\in\operatorname{\mathbb{N}}$, then $|\bigcup_{n}X_{n}|\leq\kappa$.

  * ii

Suppose $Y$ is a metric space such that for every $\epsilon>0$, there exists a
set of points $F\subseteq Y$ satisfying $|F|<\kappa$ and $Y=\cup_{x\in
F}B(x,\epsilon)$ (where $B(x,\epsilon):=\\{y\in Y:d(x,y)<\epsilon\\}$). Then
$Y$ satisfies the $\kappa^{\leq}$-chain condition.

If we further assume that $\kappa$ has uncountable cofinality (i.e. every
unbounded subset is uncountable), we get the following.

######  Theorem 2.3.

If $\kappa$ is a well-ordered cardinal with uncountable cofinality then the
following are equivalent.

  * i

If $X=(X_{1},X_{2},...)$ is a sequence of sets such that $|X_{n}|<\kappa$ for
all $n\in\operatorname{\mathbb{N}}$, then $|\bigcup_{n}X_{n}|<\kappa$.

  * ii

Suppose $Y$ is a metric space such that for every $\epsilon>0$, there exist a
set of points $F\subseteq Y$ such that $|F|<\kappa$ and $Y=\cup_{x\in
F}B(x,\epsilon)$. Then $Y$ satisfies the $\kappa$-chain condition.

The proofs of these two theorems are sufficiently similar to be presented
simultaneously.

###### Proof of 2.2 and 2.3.

($\Rightarrow$) Assume i holds. Let $Y$ be a metric space satisfying the
hypothesis of ii. Let $\mathcal{C}$ be a collection of pairwise disjoint non-
empty open subsets of $Y$, and let

$\displaystyle\mathcal{C}_{n}=\\{U\in\mathcal{C}:\exists x\in U\text{ such
that }B(x,1/n)\subseteq U\\}.$

Clearly $\mathcal{C}=\bigcup_{n}\mathcal{C}_{n}$. To see that
$|\mathcal{C}_{n}|<\kappa$, let $F\subseteq Y$ be such that $|F|<\kappa$ and
$Y=\cup_{y\in F}B(y,1/n)$. Endow $F$ with a well-order. By the construction of
$F$, the set $F\cap U$ is non-empty for all $U\in\mathcal{C}_{n}$. Let
$j:\mathcal{C}_{n}\rightarrow F$ be the map defined by letting $j(U)\in F$ be
the minimal element satisfying $j(U)\in U$. Then $j$ is an injection because
the members of $\mathcal{C}_{n}$ are pairwise disjoint. So
$|\mathcal{C}_{n}|<\kappa$. By i,
$|\mathcal{C}|=|\bigcup_{n}\mathcal{C}_{n}|\leq\kappa$ (if $\kappa$ has
uncountable cofinality, this becomes $|\mathcal{C}|<\kappa$), as desired.

($\Leftarrow$) Assume ii holds. Let $X=(X_{1},X_{2},...)$ be as in the
hypothesis of i. Assume without loss of generality that the $X_{i}$’s are
pairwise disjoint, and that $0\notin\bigcup_{n}X_{n}$. Let
$Y_{n}=X_{n}\cup\\{0\\}$ with the discrete metric, and let
$Y=\prod_{n\in\operatorname{\mathbb{N}}}Y_{n}$ with the product metric
$d(\overline{x},\overline{y})=\sum_{i=1}^{\infty}2^{-i}d_{i}(x_{i},y_{i})$. To
see that $Y$ satisfies the hypothesis of ii, let $\epsilon>0$ and choose
$n\in\operatorname{\mathbb{N}}$ such that $2^{-n}<\epsilon$. Let
$F=\\{\overline{x}\in Y:\forall i>n,x_{i}=0\\}$. Then the set of open
$\epsilon$-balls with centers in $F$ covers $X$. Furthermore,
$|F|=|Y_{1}\times...\times Y_{n}|$. But $Y_{i}<\kappa$ for all
$i\in\operatorname{\mathbb{N}}$, and $\kappa$ is an infinite well-ordered
cardinal. Therefore, it follows from basic set-theoretic considerations
(namely the fact that for infinite well-ordered cardinals $\xi$ and $\eta$ we
have $\xi+\eta=\xi\eta=\max(\xi,\eta)$) that $|Y_{1}\times...\times
Y_{n}|<\kappa$. Ergo, $Y$ satisfies the hypothesis of ii.

To see that $|\bigcup_{n}X_{n}|\leq\kappa$ (or $|\bigcup_{n}X_{n}|<\kappa$ in
the case that $\kappa$ has uncountable cofinality), for $x\in X_{n}$, let
$U_{x}=\\{0\\}^{n-1}\times\\{x\\}\times\prod_{i>n}Y_{i}.$ Then
$\mathcal{C}=\\{U_{x}:x\in\bigcup_{n}X_{n}\\}$ is a collection of pairwise
disjoint open sets, so $|\mathcal{C}|\leq\kappa$ by ii (or
$|\mathcal{C}|<\kappa$ if $\kappa$ has uncountable cofinality). But
$\mathcal{C}$ is in one-to-one correspondence with $\bigcup_{n}X_{n}$ via
$U_{x}\leftrightarrow x$. ∎

Recall that a metric space $X$ is said to be totally bounded if for every
$\epsilon>0$, $X$ can be covered by finitely many open $\epsilon$-balls.
Taking $\kappa=\aleph_{0}$, Assertion ii of 2.2 takes the form of a well-known
ZF+AC theorem on totally bounded metric spaces. The following corollary (much
in the spirit of P. Howard’s Proposition 2.7 [B1]) shows how this theorem may
be viewed as a choice principle.

######  Corollary 2.4.

The following are equivalent.

  * i

The union of a countable collection of finite sets is countable.

  * ii

Every totally bounded metric space is ccc.

2.2, 2.3, and 2.4 will serve as our prototypical examples in Section 3 as we
generalize their proof to every class of partial orders satisfying certain
conditions.

##  3\. The Categorical Perspective

Given a partially ordered set (aka a poset) $P$, we say two elements $x,y\in
P$ are incompatible if they do not share a common lower bound (aside from a
lower bound for all of $P$, if it exists). An _antichain_ is a subset
$\mathcal{C}\subseteq P$ consisting of pairwise incompatible elements. Given a
cardinal $\kappa$, we can generalize the chain conditions defined for
topological spaces to any partial order by stating that $P$ is $\kappa$-cc if
every collection of pairwise incompatible elements has cardinality $<\kappa$.
The $\kappa^{\leq}$-chain condition is similarly generalized, with the
$\aleph_{0}^{\leq}$-chain condition again being referred to interchangeably as
the countable chain condition.

The following is a generalization of a notion introduced in [horn1948measures]
that will be discussed further at the end of the present section.

######  Definition 3.1.

A poset $P$ is said to satisfy the $\sigma^{\kappa}$-chain condition if there
exists a countable collection of $\kappa$-cc subsets $P_{1},P_{2},...\subseteq
P$ such that $P=\bigcup_{n}P_{n}$. The case $\kappa=\aleph_{0}$ is called the
$\sigma$-finite chain condition.

We will say that a cardinal $\kappa$ (not necessarily well-ordered) has
_uncountable cofinality_ if a set of cardinality $\kappa$ cannot be covered by
a countable family of sets of cardinality $<\kappa$ each. (This definition
clearly does not depend on the choice of set of cardinality $\kappa$.) If
$\kappa$ is well-ordered, this condition is equivalent to every unbounded
subset of $\kappa$ being uncountable.

The starting point for Theorem 3.2 below is the following simple observation.
If the union of every family of $\aleph_{0}$ sets of cardinality $<\kappa$
each has cardinality $\leq\kappa$, then every poset satisfying the
$\sigma^{\kappa}$ chain condition must also satisfy the $\kappa^{\leq}$ chain
condition. If, in addition, $\kappa$ has uncountable cofinality, then every
poset satisfying the $\sigma^{\kappa}$ chain condition must also satisfy the
$\kappa$ chain condition.

######  Theorem 3.2.

Let $\kappa$ be a cardinal (not necessarily well-ordered or infinite), and let
$\mathfrak{A}_{\kappa}$ denote the proper class of all sequences of pairwise
disjoint non-empty sets $X=(X_{1},X_{2},...)$ such that $|X_{n}|<\kappa$ for
all $n\in\operatorname{\mathbb{N}}$. Suppose that $\mathfrak{B}$ is a family
of posets which satisfy the $\sigma^{\kappa}$ chain condition. Further suppose
that for every $X\in\mathfrak{A}_{\kappa}$, there is some
$\mathcal{F}(X)\in\mathfrak{B}$ and an injection
$f\colon\bigcup_{n}X_{n}\to\mathcal{F}(X)$ whose range is an antichain.

Then every poset in $\mathfrak{B}$ is $\kappa^{\leq}$-cc if and only if given
any countable collection of sets, each of cardinality $<\kappa$, their union
has cardinality $\leq\kappa$.

If, in addition, $\kappa$ has uncountable cofinality, then every poset in
$\mathfrak{B}$ is $\kappa$-cc if and only if given any countable collection of
sets, each of cardinality $<\kappa$, their union has cardinality $<\kappa$.

To shed some light on Theorem 3.2, note that Corollary 2.4 corresponds to the
special case where $\kappa=\aleph_{0}$ and $\mathfrak{B}$ is the proper class
of all topologies (denoted $\tau(X)$) of totally bounded metric spaces $X$,
each partially ordered by set inclusion. In this case, $\mathcal{F}$ maps a
sequence $X=(X_{1},X_{2},...)$ to $\tau(\prod_{n}(X_{n}\cup\\{X_{n}\\}))$, and
$f$ maps $x\in X_{n}$ to
$(\prod_{i<n}\\{X_{n}\\})\times\\{x\\}\times(\prod_{i>n}(X_{n}\cup\\{X_{n}\\}))$
in $\tau(\prod_{n}(X_{n}\cup\\{X_{n}\\}))=\mathcal{F}(X)$.

###### Proof of Theorem 3.2.

To prove the nontrivial implication, note that if $Y=(Y_{1},Y_{2},\dots)$ is a
sequence of sets such that $|Y_{n}|<\kappa$ for all $n$, then
$X_{n}=Y_{n}\setminus\bigcup_{j<n}Y_{j}$ defines an element of
$\mathfrak{A}_{\kappa}$ such that $\bigcup_{n}X_{n}=\bigcup_{n}Y_{n}$. It
therefore suffices to consider elements of $\mathfrak{A}_{\kappa}$. Let
$X=(X_{1},X_{2},...)\in\mathfrak{A}_{\kappa}$ and let
$f:\bigcup_{n}X_{n}\rightarrow\mathcal{F}(X)$ be an injection whose image is
an antichain. Since $\mathcal{F}(X)$ is $\kappa^{\leq}$-cc, we have
$|f(\bigcup_{n}X_{n})|\leq\kappa$. But $f(\bigcup_{n}X_{n})$ is in bijection
with $\bigcup_{n}X_{n}$, so $|\bigcup_{n}X_{n}|\leq\kappa$, as desired.

If $\kappa$ has uncountable cofinality, then for every
$X\in\mathfrak{A}_{\kappa}$ we have $|\bigcup_{n}X_{n}|\leq\kappa$ if and only
if $|\bigcup_{n}X_{n}|<\kappa$, and the desired equivalence reduces to the one
proven for an arbitrary $\kappa$. ∎

A typical example of a poset satisfying the $\sigma$-finite chain condition is
the set of nonzero elements of a probability measure algebra. Setting $P_{n}$
to be the family of all sets of measure $>1/n$, for $n\geq 2$, it is clear
that $P_{n}$ cannot contain an antichain of cardinality $n$. Since the
cardinality of antichains in $P_{n}$ is uniformly bounded, we have an even
stronger chain condition, called the $\sigma$-bounded chain condition. These
two chain conditions were introduced by Horn and Tarski in the course of their
study of measure algebras [horn1948measures]. It took until
[thummel2014problem] to prove that the $\sigma$-bounded chain condition is
stronger than the $\sigma$-finite chain condition. While the $\sigma$-finite
chain condition implies the countable chain condition in the presence of the
Axiom of Choice, we have the following corollary of Theorem 3.2, obtained by
setting $\kappa=\aleph_{0}$.

######  Corollary 3.3.

The following are equivalent.

  * i

The union of a countable collection of finite sets is countable.

  * ii

Every poset satisfying the $\sigma$-finite chain condition is ccc.

  * iii

Every poset satisfying the $\sigma$-bounded chain condition is ccc.

The rest of this paper is focused on deriving corollaries of 3.2 in the
context of probability theory and Hilbert spaces.

##  4\. Probability

Given a Boolean algebra $\textbf{B}=(B,\textbf{0},\textbf{1},\wedge,\vee)$ and
elements $x,y\in B$, we write $x\leq y$ if $x\wedge y=x$. In this way, $B$
admits a canonical poset structure. B is said to be ccc if the poset $B$ is
ccc. Note that elements $x,y\in B$ are incompatible in the sense of posets if
and only if $x\wedge y=\textbf{0}$.

A _probability measure algebra_ is a pair $(\textbf{B},\mu)$, where
$\textbf{B}=(B,\textbf{0},\textbf{1},\wedge,\vee)$ is a Boolean algebra and
$\mu:B\rightarrow[0,1]$ is a finitely additive probability measure (i.e.
$\mu(\textbf{1})=1$) assigning a positive measure to all non-zero elements.
Note that we do not require B to be complete. (Recall that a Boolean algebra
is _complete_ if every nonempty ßsubset has a supremum.) Given a probability
algebra $(\textbf{B}=(B,\textbf{0},\textbf{1},\wedge,\vee),\mu)$, we define
$s(\textbf{B},\mu)=B$; this is a forgetful functor from the category of all
probability algebras into the category of Boolean algebras. As previously
mentioned, when AC is at our disposal, every probability algebra is ccc. The
following theorem makes the role of AC much more precise.

######  Theorem 4.1.

The following are equivalent.

  * i

The union of a countable collection of finite sets is countable.

  * ii

Every probability algebra is ccc.

###### Proof.

We employ the framework of 3.2 with $\kappa=\aleph_{0}$, and where
$\mathfrak{B}$ is the proper class of all posets $s(\textbf{B},\mu)$ (for
$(\textbf{B},\mu)$ ranging over all probability algebras).

($\Rightarrow$) Let $B=s(\textbf{B},\mu)\in\mathfrak{B}$ and let
$\mathcal{C}\subseteq B$ be an antichain. Without loss of generality, assume
$\textbf{0}\notin\mathcal{C}$. Let

$\displaystyle\mathcal{C}_{n}=\bigg{\\{}x\in\mathcal{C}:\mu(x)>\frac{1}{n}\bigg{\\}}$

so that $\mathcal{C}=\bigcup_{n}\mathcal{C}_{n}$. By additivity of $\mu$, we
have $|\mathcal{C}_{n}|\leq n$ for all $n$, so $\mathcal{C}$ is a union of
countably many finite sets. Thus i implies ii, as desired.

($\Leftarrow$) For this direction, we need to construct the map
$\mathcal{F}:\mathfrak{A}_{\aleph_{0}}\rightarrow\mathfrak{B}$ required by
3.2. Given a sequence of pairwise disjoint non-empty finite sets
$X=(X_{1},X_{2},...)\in\mathfrak{A}_{\aleph_{0}}$, define $\textbf{B}_{X}$ to
be the Boolean algebra of all subsets of $\bigcup_{n}X_{n}$. For $a\in X_{n}$,
define $\mu_{X}(\\{a\\})=|X_{n}|^{-1}2^{-n}$, and for an arbitrary subset
$S\subseteq\bigcup_{n}X_{n}$, define

$\displaystyle\mu_{X}(S)=\sup_{F\subseteq S\text{ finite}}\sum_{a\in
F}\mu_{X}(\\{a\\}).$

Then we define $\mathcal{F}$ by $\mathcal{F}(X)=s(\textbf{B}_{X},\mu_{X})$. To
satisfy the hypothesis of Theorem 3.2, define
$f:\bigcup_{n}X_{n}\rightarrow\mathcal{F}(X)$ by mapping
$a\in\bigcup_{n}X_{n}$ to the singleton $\\{a\\}\in\mathcal{F}(X)$. By 3.2, we
conclude that ii implies i, as desired. ∎

##  5\. Hilbert Spaces

In the presence of Countable Choice (that is, the assertion that every
countable collection of nonempty sets admits a choice function, denoted
$\text{AC}_{\omega}$), a metric space is Cauchy complete (i.e. every Cauchy
sequence converges) if and only if every nested sequence of non-empty closed
sets with vanishing diameter intersects in a singleton. Ergo, when working
with $\text{AC}_{\omega}$, it is customary to generically call a metric space
satisfying one of these two conditions "complete." However, in the absence of
any form of choice, Cauchy completeness becomes a weaker condition. We follow
[A1] and say that a metric space in which every nested sequence of non-empty
closed sets with vanishing diameter intersects in a singleton is _$\sigma$
-complete_ (some authors use the term _Cantor complete_). As demonstrated in
Section 2 of [A1], an inner product space which is $\sigma$-complete retains
the fundamental properties of a Hilbert space such as the closest vector
property, the existence of orthogonal complements, and the Riesz
representation theorem. We follow [A1] (Definition 1.0.1.ii) and say that an
inner product space is a _Hilbert space_ if it is $\sigma$-complete with
respect to the induced norm.

It is straightforward to verify that, the theory of orthogonal projections
being unchanged in the absence of choice, we can still obtain Bessel’s
inequality. Hilbert spaces and operators on Hilbert spaces in the absence of
AC have been studied in [BrunnerLinear] and [BrunnerSB] from a slightly
different point of view; see the discussion in 6.1.2 of [A1].

Given a set $X$, we define $\ell^{2}(X)$ to be the set of all maps
$f:X\rightarrow\operatorname{\mathbb{C}}$ such that $\sum_{x\in
X}|f(x)|^{2}<\infty$ ([A1], Definition 3.0.2). By equipping $\ell^{2}(X)$ with
the obvious inner product ([A1], 3.0.3), one shows that $\ell^{2}(X)$ is a
Hilbert space ([A1], Proposition 3.0.4) and that the collection of
characteristic maps $\zeta_{x}:X\rightarrow\operatorname{\mathbb{C}}$ of
singletons $\\{x\\}\subseteq X$ forms an orthonormal basis ([A1], 3.0.5). In
fact, Hilbert spaces constructed this way are the canonical Hilbert spaces
with orthonormal basis, in the sense that given any Hilbert space
$\mathcal{H}$ with orthonormal basis $B$, the map
$B\ni\vec{x}\mapsto\zeta_{\vec{x}}\in\ell^{2}(B)$ extends to an isometric
linear isomorphism $\mathcal{H}\rightarrow\ell^{2}(B)$ ([A1], 3.0.6).

######  Theorem 5.1.

The following are equivalent.

  * i

The union of a countable collection of finite sets is countable.

  * ii

If $\mathcal{H}$ is a Hilbert space, $B\subseteq\mathcal{H}$ is an orthonormal
subset, and $\vec{x}\in\mathcal{H}$, then the set

$\displaystyle T:=\\{\vec{e}\in B:\langle\vec{x},\vec{e}\rangle\neq 0\\}$

is countable.

When AC is at our disposal, ii is easily obtained as a theorem. 5.1 makes
precise the extent to which the Axiom of Choice is needed to prove this
theorem.

###### Proof of 5.1.

($\Rightarrow$) Assume i holds. By 4.1, every probability algebra is ccc. Let
$\mathcal{H}$ be a Hilbert space, let $\vec{x}\in\mathcal{H}$, and let
$B\subseteq\mathcal{H}$ be an orthonormal subset. Let

$\displaystyle N=\sup_{F\subseteq B\text{ finite}}\sum_{\vec{e}\in
F}|\langle\vec{x},\vec{e}\rangle|^{2}.$

Note that $N\leq||\vec{x}||^{2}<\infty$ by Bessel’s inequality. If $N=0$, then
clearly $T$ is empty and we are done. So assume $N\neq 0$. This allows us to
turn the powerset of $T$ into a probability algebra by defining

$\displaystyle\mu(S)=\frac{1}{N}\sup_{F\subseteq S\text{
finite}}\sum_{\vec{e}\in F}|\langle\vec{x},\vec{e}\rangle|^{2}$

for all $S\subseteq T$. By 4.1, we conclude that the collection of singletons
$\\{\\{\vec{e}\\}:\vec{e}\in T\\}$ is countable, as desired.

($\Leftarrow$) For this direction, we employ the framework of 3.2 in the case
$\kappa=\aleph_{0}$. Given a triple $(\mathcal{H},\vec{x},B)$, where
$\mathcal{H}$ is a Hilbert space, $\vec{x}\in\mathcal{H}$ is a vector, and
$B\subseteq\mathcal{H}$ is an orthonormal subset, define

$o(\mathcal{H},\vec{x},B)=\text{PowerSet}\\{\vec{e}\in
B:\langle\vec{x},\vec{e}\rangle\neq 0\\},$

ordered by set inclusion. We take $\mathfrak{B}$ to be the proper class of all
posets $o(\mathcal{H},\vec{x},B)$ (where $(\mathcal{H},\vec{x},B)$ ranges over
all triples). Note that Assertion ii of 5.1 is equivalent to the assertion
that every member of $\mathfrak{B}$ is ccc. Next, we must construct a suitable
map $\mathcal{F}:\mathfrak{A}_{\aleph_{0}}\rightarrow\mathfrak{B}$. Given
$Y=(Y_{1},Y_{2},...)\in\mathfrak{A}_{\aleph_{0}}$, we define
$\mathcal{F}(Y)=o(\ell^{2}(\bigcup_{n}Y_{n}),\vec{x}_{Y},B_{Y})$, where
$\vec{x}_{Y}\in\ell^{2}(\bigcup_{n}Y_{n})$ is defined by $Y_{n}\ni
a\mapsto|Y_{n}|^{-1}n^{-1}=\vec{x}_{Y}(a)\in\operatorname{\mathbb{C}}$, and
$B_{Y}$ is the standard orthonormal basis for $\ell^{2}(\bigcup_{n}Y_{n})$. To
see that the hypothesis of Theorem 3.2 is satisfied, define the map
$f:\bigcup_{n}Y_{n}\rightarrow\mathcal{F}(X)$ by $Y_{n}\ni
a\mapsto\\{\zeta_{a}\\}=f(a)\in\mathcal{F}(X)$. ∎

Example 6.3.1 of [A1] shows that the failure of i gives the existence of a
Hilbert space with two orthonormal bases of different cardinalities. It is not
clear whether it also implies the existence of a Hilbert space with no
orthonormal basis (6.1.2 of [A2]; see however Theorem 6.2.3 of [A1], where
such a space has been constructed from a particularly strong failure of i).

##  6\. Conclusion

We see that chain conditions are closely related to the cardinality of
countable unions. The ZF+AC proof that Cauchy complete metric spaces are
$\sigma$-complete makes straightforward, seemingly indispensable use of the
Axiom of Countable Choice ($\text{AC}_{\omega}$). It is well-known that the
equivalence of sequential and topological continuity in the context of metric
spaces implies $\text{AC}_{\omega}$. This suggests that theorems about
sequential convergence in metric spaces are a good place to look for
equivalents of $\text{AC}_{\omega}$. Moreover, the equivalence of the Baire
Category Theorem on Cauchy complete metric spaces (abbreviated BCT) and the
Axiom of Dependent Choice demonstrates that theorems about Cauchy complete
metric spaces may imply rather strong forms of choice. We are lead to the
following.

######  Question 6.1.

Can we use the equivalence of Cauchy completeness and $\sigma$-completeness in
the context of metric spaces to derive the Axiom of Countable Choice?

Conversely, a typical ZF+AC proof that ccc metric spaces are second-countable
(i.e. admit a countable base) uses Zorn’s Lemma for posets in which every
well-ordered chain is countable. This weak form of Zorn’s Lemma (which we call
$Z_{\aleph_{1}}$) turns out to be equivalent to a strong form of Dependent
Choice, which, in turn, implies BCT. Given the vastness of the class of Cauchy
complete metric spaces, it seems unlikely that a theorem about ccc metric
spaces could be used to prove a theorem about Cauchy complete metric spaces.
After all, unlike ccc metric spaces, Cauchy complete metric spaces may admit
any number of pairwise disjoint non-empty open sets. Ergo, it is improbable
that the arrow may be reversed in this case: using the fact that ccc metric
spaces are second-countable to prove $Z_{\aleph_{1}}$. In the interest of
viewing this property of ccc metric spaces as a choice principle, we are lead
to the following.

######  Question 6.2.

Can the equivalence of the countable chain condition and second-countability
in the context of metric spaces be proven using a weaker form of choice than
$Z_{\aleph_{1}}$?

In [A1] it is asked whether every Hilbert space is a subspace of a Hilbert
space admitting an orthonormal basis (in the absence of any form of choice). A
good place to start is by looking at spaces of the form
$L^{2}(X,\mathcal{A},\mu)$ (for measure spaces $(X,\mathcal{A},\mu)$), as this
is an abundant source of everyday Hilbert spaces. To simplify matters even
further, take $(X,\mathcal{A},\mu)$ to be a measure algebra (i.e.
$\mathcal{A}$ is a Boolean algebra and $\mu$ is a finitely additive measure).

######  Question 6.3.

Is every space of the form $L^{2}(X,\mathcal{A},\mu)$ (for
$(X,\mathcal{A},\mu)$ a measure algebra) a subspace of a Hilbert space
admitting an orthonormal base?

We conclude with a question about the nature of the specific failure of choice
used in this paper. Recall that a set $X$ is Dedekind-finite if there is no
injection from $\operatorname{\mathbb{N}}$ into $X$. If there is a Dedekind-
finite set $X$ such that the family of all finite subsets of $X$ is not
Dedekind-finite, then a simple argument shows that $X$ contains the union of
an infinite family of disjoint, nonempty, finite sets. The union of this
family is infinite and Dedekind-finite, and therefore not countable.

######  Question 6.4.

Assume that the union of any countable collection of finite sets is countable.
Can one prove that there is a Dedekind-finite set $X$ such that the set of all
finite subsets of $X$ is not Dedekind-finite?

## References

  * [1]
BlackadarB.FarahI.KaragilaA.Hilbert spaces without countable acarXiv preprint
arXiv:2304.096022023@article{A1, author = {Blackadar, B.}, author = {Farah,
I.}, author = {Karagila, A.}, title = {Hilbert Spaces Without Countable AC},
journal = {arXiv preprint arXiv:2304.09602}, date = {2023}}

  * [3]
BlackadarB.http://wolfweb.unr.edu/homepage/bruceb/.Real
analysis(2023)@book{BlackadarReal, author = {Blackadar, B.}, note =
{\url{http://wolfweb.unr.edu/homepage/bruceb/}.}, title = {Real Analysis},
date = {(2023)}}

  * [5]
BrunnerN.SvozilK.BaazM.DocumentMathematical Logic QuarterlyISSN 0942-5616Math.
Logic Quart.03E25 (03B30 03E35 03E75 03F65 04A25 35Q40 81P10)1398140Paul E.
Howard3319–340The axiom of choice in quantum theoryLink421996https://doi-
org.unr.idm.oclc.org/10.1002/malq.19960420128https://doi.org/10.1002/malq.19960420128MathReview
(Paul E. Howard)@article{BrunnerSB, author = {Brunner, N.}, author = {Svozil,
K.}, author = {Baaz, M.}, doi = {10.1002/malq.19960420128}, fjournal =
{Mathematical Logic Quarterly}, issn = {0942-5616}, journal = {Math. Logic
Quart.}, mrclass = {03E25 (03B30 03E35 03E75 03F65 04A25 35Q40 81P10)},
mrnumber = {1398140}, mrreviewer = {Paul E. Howard}, number = {3}, pages =
{319–340}, title = {The axiom of choice in quantum theory}, url =
{https://doi-org.unr.idm.oclc.org/10.1002/malq.19960420128}, volume = {42},
year = {1996}, bdsk-url-1 = {https://doi-
org.unr.idm.oclc.org/10.1002/malq.19960420128}, bdsk-url-2 =
{https://doi.org/10.1002/malq.19960420128}}

  * [7]
BrunnerN.Mathematica JaponicaISSN 0025-5513Math. Japon.47B37 (03E25 47A99
54E52)83582711–16Linear operators and Dedekind sets311986MathReview
Entry@article{BrunnerLinear, author = {Brunner, N.}, fjournal = {Mathematica
Japonica}, issn = {0025-5513}, journal = {Math. Japon.}, mrclass = {47B37
(03E25 47A99 54E52)}, mrnumber = {835827}, number = {1}, pages = {1–16}, title
= {Linear operators and {D}edekind sets}, volume = {31}, year = {1986}}

  * [9]
De la CruzO.HallE.HowardP.KeremedisK.RubinJ.Metric spaces and the axiom of
choiceMath. Logic Quart. 492003no. 5, 455-466, DOI
10.1002/malq.200310049@article{i1, author = {De la Cruz, O.}, author = {Hall,
E.}, author = {Howard, P.}, author = {Keremedis, K.}, author = {Rubin, J.},
title = {Metric spaces and the axiom of choice}, journal = {Math. Logic Quart.
\textbf{49}}, date = {2003}, note = {no. 5, 455-466, DOI
10.1002/malq.200310049}}

  * [11]
FremlinD.Measure theory5Torres–Fremlin2005@book{A3, author = {Fremlin, D.},
title = {Measure Theory}, volume = {5}, publisher = {Torres–Fremlin}, date =
{2005}}

  * [13]
GutierresG.Sequential topological conditions in $\operatorname{\mathbb{R}}$ in
the absence of the axiom of choiceMath. Log. Quart. 492003no. 3, 293-298, DOI
10.1002/malq.200310029@article{c3, author = {Gutierres, G.}, title =
{Sequential topological conditions in $\R$ in the absence of the axiom of
choice}, journal = {Math. Log. Quart. \textbf{49}}, date = {2003}, note = {no.
3, 293-298, DOI 10.1002/malq.200310029}}

  * [15]
HornA.TarskiA.Trans. Amer. Math. Soc.3467–497Measures in boolean
algebras641948@article{horn1948measures, author = {Horn, A.}, author =
{Tarski, A.}, journal = {Trans. Amer. Math. Soc.}, number = {3}, pages =
{467–497}, title = {Measures in Boolean algebras}, volume = {64}, year =
{1948}}

  * [17]
HowardP.RubinJ.Consequences of the axiom of choiceAmerican Mathematical
Society1998@book{d1, author = {Howard, P.}, author = {Rubin, J.}, title =
{Consequences of the Axiom of Choice}, publisher = {American Mathematical
Society}, date = {1998}}

  * [19]
HowardP.TachtsisE.On metrizability and compactness of certain products without
the axiom of choiceTop. Appl.2021@article{B1, author = {Howard, P.}, author =
{Tachtsis, E.}, title = {On metrizability and compactness of certain products
without the Axiom of Choice}, journal = {Top. Appl.}, date = {2021}}

  * [21]
JechT.The axiom of choiceNorth-Holland Publishing Company1973@book{A2, author
= {Jech, T.}, title = {The Axiom of Choice}, publisher = {North-Holland
Publishing Company}, date = {1973}}

  * [23]
KaragilaA.SchlichtP.Proceedings of the Royal Society A: Mathematical,
Physical, and Engineering Sciences 476How to have more things by forgetting
how to count them2020no. 2239@article{c4, author = {Karagila, A.}, author =
{Schlicht, P.}, journal = {Proceedings of the Royal Society A: Mathematical,
Physical, and Engineering Sciences \textbf{476}}, title = {How to have more
things by forgetting how to count them}, year = {2020}, note = {no. 2239}}

  * [25]
KaragilaA.SchweberN.2022-10-11 17:38:37 -04002022-10-11 18:07:36 -0400European
J. Math.393–410Choiceless chain conditions82022@article{A6, author =
{Karagila, A.}, author = {Schweber, N.}, date-added = {2022-10-11 17:38:37
-0400}, date-modified = {2022-10-11 18:07:36 -0400}, journal = {European J.
Math.}, pages = {393–410}, title = {Choiceless Chain Conditions}, volume =
{8}, year = {2022}}

  * [27]
KeremedisK.On sequentially compact and related notions of compactness of
metric spaces in ZFBull. Pol. Acad. Sci. Math.646429–462016@article{c2, author
= {Keremedis, K.}, title = {On sequentially compact and related notions of
compactness of metric spaces in \textbf{ZF}}, journal = {Bull. Pol. Acad. Sci.
Math.64}, number = {64}, pages = {29–46}, date = {2016}}

  * [29]
KeremedisK.TachtsisE.Compact metric spaces and weak forms of the axiom of
choiceMath. Log. Quart. 47no. 1, 117-1282001@article{i3, author = {Keremedis,
K.}, author = {Tachtsis, E.}, title = {Compact Metric Spaces and Weak Forms of
the Axiom of Choice}, journal = {Math. Log. Quart. \textbf{47}}, note = {no.
1, 117-128}, date = {2001}}

  * [31]
KeremedisK.TachtsisE.Weak axioms of choice for metric spacesProc. Amer. Math.
Soc. 1332005no. 12, 3691–3701@article{i2, author = {Keremedis, K.}, author =
{Tachtsis, E.}, title = {Weak Axioms of Choice for Metric Spaces}, journal =
{Proc. Amer. Math. Soc. \textbf{133}}, date = {2005}, note = {no. 12,
3691–3701}}

  * [33]
KeremedisK.WajchE.Denumerable cellular families in hausdorff spaces and towers
of boolean algebras in ZFarXiv preprint arXiv:2001.006192020@article{c1,
author = {Keremedis, K.}, author = {Wajch, E.}, title = {Denumerable cellular
families in Hausdorff spaces and towers of Boolean algebras in \textbf{ZF}},
journal = {arXiv preprint arXiv:2001.00619}, date = {2020}}

  * [35]
KunenK.Set theory: an introduction to independence proofsElsevier Science
Publishers B.V.1980@book{A5, author = {Kunen, K.}, title = {Set Theory: An
Introduction to Independence Proofs}, publisher = {Elsevier Science Publishers
B.V.}, date = {1980}}

  * [37]
MooreG.Zermelo’s axiom of choice: its origins, development, and
influenceSprifnger-Verlag1982@book{A4, author = {Moore, G.}, title =
{Zermelo's Axiom of Choice: Its Origins, Development, and Influence},
publisher = {Spriƒnger-Verlag}, date = {1982}}

  * [39]
ThümmelE.Proc. Amer. Math. Soc.61997–2000The problem of horn and
tarski1422014@article{thummel2014problem, author = {Th{\"u}mmel, E.}, journal
= {Proc. Amer. Math. Soc.}, number = {6}, pages = {1997–2000}, title = {The
problem of Horn and Tarski}, volume = {142}, year = {2014}}

  * [41]