Characterization of paracompactness

[quasar987]

A topological space X may be defined as paracompact by the condition that every open cover U of X admits a refinement U' such that every point of X intersects only a finite number of elements of U'.

A seemingly stronger condition on X would be that every open cover U of X admits a refinement U' such that around every point x of X, there is an open nbhd A which intersects only finitely many elements of U'.

I'm pretty sure that in fact these conditions are equivalent (at least for metric spaces) but I'm having trouble proving it.

 

[Billy Bob]

I thought your second paragraph was the definition of paracompact. The first paragraph would be metacompact. Paracompactness implies metacompactness obviously. The converse is not true, and not obvious at all. Every metric space is paracompact, hence metacompact. Hopefully the additional term will give you something to search on. If I were Mary Ellen Rudin, I could prove these myself. But I'm not, so I'd look them up! :)

 

[quasar987]

Ok, thanks Billy bob!

 

[Billy Bob]

P.S.

I looked it up in Steen and Seebach. Here are two examples that are not excruciatingly difficult.

First is #54 Interlocking Interval Topology, where X = positive reals excluding positive integers. Base consists of the sets S_n, where S_n is the union of (0,1/n) and (n,n+1) where n=positive integer. The open cover {S_n} has no open refinement. Consider S_1 to see that X is not paracompact. On the other hand, metacompactness is not hard to show.

#54 is not Hausdorff. If you want Hausdorff, look at second example.

Second example is #64 Smirnov's Deleted Sequence Topology, which apparently the same as what Munkres (2nd edition) calls the K-topolgy. Let X=real line. Let K={1/n : n is a positive integer}. Let scriptB = usual open intervals, and let scriptB_K be scriptB union {sets of the form B - K where B is in scriptB}. Then scriptB_K is a basis for the K-topology. It looks like K-open sets are of the form U-L where U is usual open and L is a subset of K. Then the K-topology could be shown to be metacompact. It is not paracompact: consider for each positive integer n the set O_n = (reals-K) union {1/n}, and consider covering by {O_n}.

 

[cogito²]

I just take the definition of paracompact to mean there exists a partition of unity...

 

[quasar987]

Apparently, that equivalence only holds for Hausdorff spaces cogito²: http://en.wikipedia.org/wiki/Paracompact#Partitions_of_unity.

 

[cogito²]

Apparently, that equivalence only holds for Hausdorff spaces cogito²: http://en.wikipedia.org/wiki/Paracompact#Partitions_of_unity.

Yeah well I'm not sure the last time I considered a partition of unity on a non-Hausdorff space so I'll probably be okay...