TITLE: Topological space M with partition of unity--->M paracompact. John Lee Problems
QUESTION [9 upvotes]: Suppose $M$ 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.

REPLY [3 votes]: $X=\lbrace X_a \rbrace$ and let $\lbrace P_a\rbrace$ be the associated partition of unity, the index set for both is $A$. Define $A'=\lbrace a|a\in A,P_a\neq 0\rbrace$ and finally define $V_a=P_a^{-1}(0,1]$ for $a\in A'$, open in $M$. It is easy to see that $\lbrace V_a|a\in A'\rbrace$ is a locally finite refinement.