Theorem Suggestion
If a space is:
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?
Theorem Suggestion
If a space is:
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 + ~ParacompactProof/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:
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?