Help with understanding the concept of paracompactness. The definition for paracompactness that I'm given is as follows:
"A set $U\subseteq\epsilon$ is paracompact with respect to the topology ($\epsilon$,$\tau$) if and only if every open cover {$O_i$} of $U$ has a locally finite refinement."
However I am having a hard time understanding what locally finite refinement means. I've looked on wikipedia and various other websites but it doesn't seem to do it for me. Could anyone please explain paracompactness in a simple sort of fashion with emphasis of the definition of locally finite refinement?
Examples and graphs could also help - Cheers :)
 A: Note first that the definition of compactness [sic] can be viewed as an adversarial game: You start with a set $K$ in some topological space. Your opponent picks an arbitrary family of open sets whose union contains $K$. Your task is to "discard" all but finitely many of your opponent's sets in a way that the remaining sets still cover $K$. To say $K$ is compact is to say "you have a winning strategy against a perfect opponent".
Paracompactness can be give a similar interpretation. You start with a set $U$ in some topological space. Your opponent picks an arbitrary family of open sets whose union contains $U$. Your task is a bit more elaborate than before:


*

*You're allowed to "shrink" the sets your opponent gives you, i.e., to replace any open set by an open subset ("refinement").

*The subsets you pick sets must cover $U$, but every point of $U$ must have a neighborhood meeting only finitely many of your sets ("locally finite").
To say $U$ is "paracompact" means you have a winning strategy against a perfect opponent.
A: An open cover $\{U_i\}$ is a refinement of an open cover $\{V_j\}$ if for each $i$ there exists a $j$ with $U_i \subseteq V_j$.
An open cover $\{U_i\}$ of $X$ is locally finite, if for every $x\in X$, there exists an open neighborhood of $x$ which only intersects finitely many of the $\{U_i\}$.
