Theorem Suggestion: Locally compact + Has a group topology => Paracompact

[GeoffreySangston]

Theorem Suggestion

If a space is:

• Locally compact P130
• Has a group topology P87

then it is Paracompact P30

Rationale

This theorem would demonstrate that no spaces satisfy the following search:

π-Base, Search for Locally compact + Has a group topology + ~Paracompact

Proof/References

Theorem 8.13 of Hewitt-Ross (Volume 1) states that a locally compact $T_0$ group is paracompact, hence normal. I don't think $T_0$ is actually used in the proof. Note Hewitt-Ross includes the following on page 9:

In agreement with many writers on set-theoretic topology, although not with KELLEY [2], we make T0 separation part of the axioms of regularity and complete regularity. Thus in our terminology, regular and completely regular spaces satisfy Hausdorff's separation axiom.

So I assume $T_0$ is just added for the "hence normal" part. I checked page 113 of Kelley, which defines regular without $T_0$. Anyways, the proof from Hewitt-Ross is (more or less verbatim) repeated here. (Formatting the latex with the weird mathjax constraints wasn't working, so to read it, please go to edit and then copy and paste the following into the mardown box on the preview page: https://topology.pi-base.org/dev/preview.)

Let $G$ be a locally compact group. Suppose $U \subset G$ is a symmetric neighborhood of the identity which is precompact. It is straightforward that $L := \bigcup_{n = 1}^\infty U^n$ is an open subgroup of $G$. Since every open subgroup of a topological group is also closed, $L$ is closed. And since $\overline{U} \subset U^2$, $L$ is {P17}. Then by {T122}, $L$ is {P18}. Let $\mathcal{V}$ denote an open cover of $G$. For each coset $xL \subset G$, {P18} implies there is a countably subfamily ${V_{xL}^n}{n = 1}^\infty$ of $\mathcal{V}$ such that $x L \subset \bigcup{n = 1}^\infty V_{xL}^n$. For each $n$, define a family $\mathcal{W}n := {V{xL}^n \cap (xL) : xL \in G/L}$. The union of these families $\mathcal{W} = \bigcup_{n = 1}^\infty \mathcal{W}_n$ is an open $\sigma$-locally finite refinement of $\mathcal{V}$. Theorem 28 on page 156 of Kelley states that for regular spaces, paracompactness is equivalent to the assertion that every open cover has an open $\sigma$-locally finite refinement. Since a topological group is regular (Explore), it follows that $G$ is {P30}.

Maybe this would be a bit shorter if the equivalences from Kelley, or other sources, were added to the paracompact space (Edit: well, maybe this is not a great idea since it assumes regular...)? Henno Brandsma actually states here that (for connected, though this probably doesn't matter) locally compact groups are strongly paracompact. Does a simple modification of the above give that?

[prabau]

Please double check me on the following.

The following properties hold for a space $X$ iff they hold for the Kolmogorov quotient of $X$:

• normal
• group topology (the set of points topol. indistinguishable from the unit element $e$ of the group is the closure of $e$, which is a normal subgroup (the closure of a normal subgroup is normal), and the quotient group corresponds to the Kolm. quotient with quotient topology)
• paracompact (without separation axiom)
• locally compact (I double checked and it seems correct)
• weakly locally compact

So if need be, we can assume T0, equivalent to Tychonoff.

[GeoffreySangston]

@prabau Will edit this comment with my response tomorrow. (I'm trying to limit my internet to 1 hour a day as a new years resolution, and the post used my hour haha.)

Edit: Okay well I checked the properties and they're all preserved by Kolmogorov quotients and vice versa. So we could mention $G$ is paracompact if and only its $T_0$ quotient is, and then cite Hewitt-Ross. Though I see improvements to what's in Hewitt-Ross can be made by modifying the argument under the hood.

[prabau]

Good resolution :-)

[prabau]

One more background comment. Topological group are regular (even completely regular). And for regular spaces, weakly locally compact, locally compact, and locally relatively compact are all equivalent. So it's enough that the unit element of the group has a single compact nbhd, and the whole group is locally compact.

[prabau]

It seems the argument in your proof can be simplified a little.
First construct the clopen subgroup $L$. It's Lindelof + regular, hence fully normal.
The group $G$ is the topological sum of all the cosets of $L$.

Meta-property: Fully normal is preseved by arbitrary topological sums (or is it better to say coproduct?)
(Easy to see directly, similar to the proof of the same meta-property for paracompactness, Engelking 5.1.30)

Hence $G$ is fully normal.

[Moniker1998]

Lindelöf regular space is (strongly) paracompact.

This is something that doesn't exist on pi-base https://topology.pi-base.org/spaces?q=regular+%2B+Lindelof+%2B+hausdorff+%2B+not+strongly+para

[yhx-12243]

Lindelöf regular space is (strongly) paracompact.

This is something that doesn't exist on pi-base https://topology.pi-base.org/spaces?q=regular+%2B+Lindelof+%2B+hausdorff+%2B+not+strongly+para

This is pending #992.