Partitions of unity question

From: Tony (Ttiger222_at_hotmail.com)
Date: 01/31/05 

Date: Mon, 31 Jan 2005 17:24:01 -0500

If M is a topological space with the property that for every open cover X of
M, there exists a partition of unity subordinate to X, show that M is
paracompact.

I can't quite see how to do this. If {f_i} is a partition of unity for X =
{X_i}, then certainly

X = Union of {supp(f_i)} (over i), since if x is in M, then there exists a
function f_k such that f_k(x) is not zero since by definition of parition of
unity,

sum f_i(x) = 1 for all x in M (where the sum is taken over all i).

So if supp(f_i) was open for all i, then {supp(f_i)} would be a locally
finite refinement of {X_i} since supp(f_i) is a subset of X_i and the supp's
form a locally finite set.

But they may not be open. So I'm not sure what to do. For a particular k,
can I maybe write supp(f_k) as a union of a finite amount of open sets?
Because it seems clear to me that a finite refinement of a locally finite
set is locally finite. But it doesn't appear clear to me that any
refinement of a locally finite set is locally finite.

Any help is highly appreciated, Thank you,

Tony

Relevant Pages

  • Re: Partitions of unity proof roadblock
    ... to partitions of unity. ... E admits C^k partitions of unity subordinate to any open covering ... refine again to a locally finite open covering {V_ ... contained in U .The function g of the partition of unity below that ...
    (sci.math)
  • Re: Partitions of unity proof roadblock
    ... to partitions of unity. ... E admits C^k partitions of unity subordinate to any open covering ... refine again to a locally finite open covering {V_ ... contained in U .The function g of the partition of unity below that ...
    (sci.math)
  • Re: Partitions of unity proof roadblock
    ... to partitions of unity. ... E admits C^k partitions of unity subordinate to any open covering ... refine again to a locally finite open covering {V_ ... contained in U .The function g of the partition of unity below that ...
    (sci.math)
  • Re: Partitions of unity question
    ... >> M, there exists a partition of unity subordinate to X, show that M is ... >> Because it seems clear to me that a finite refinement of a locally finite ... be the _closure_ _of_ the set where the function is non-zero ...
    (sci.math)
  • Re: Partitions of unity question
    ... Tony wrote: ... > M, there exists a partition of unity subordinate to X, show that M is ... > Because it seems clear to me that a finite refinement of a locally finite ... > refinement of a locally finite set is locally finite. ...
    (sci.math)