Locally Compact + Fully Normal implies Strongly Paracompact

[justforphone42069-cloud]

Theorem Suggestion

If a space is:

• locally compact P130
• fully normal P34

then it is strongly paracompact P145.

Rationale

All metrizable spaces are fully normal (as are pseudometrizable spaces), so of most typical interest is that locally compact metrizable (resp. pseudometrizable) spaces are strongly paracompact.

Proof/References

https://www.sciencedirect.com/science/article/abs/pii/S0924650909700546

Nagata's General Topology

Note also that for Hausdorff spaces, fully normal is equivalent to paracompact - this is Stone's Theorem. Thus:

• locally compact P130
• paracompact P34
• $T_2$ P3

then it is strongly paracompact P145.

https://www.math.uni-bielefeld.de/~tcutler/pdf/Paracompact%20Spaces.pdf

[Moniker1998]

You didn't include where this is located. It's exercise V.8. Nagata uses Hausdorff assumption. Also different types of local compactness matter. One has to carefully check that proof goes through with these assumptions.

[Moniker1998]

https://math.stackexchange.com/questions/3246970/connected-locally-compact-paracompact-hausdorff-space-is-exhaustible-by-compac/5028016#5028016

I believe it actually should be weakly locally compact + para-Lindelof + regular => strongly paracompact

We probably can't get rid of the regularity assumption.

Why I think that we need regularity is because in the exercise they use that disjoint unions of Lindelof spaces (or more generally strongly paracompact spaces) are strongly paracompact. For this we need that Lindelof + regular => strongly paracompact.

Either that or there exist alternative proofs (although I doubt that).