# Can't understand a detail in paracompactness->normality

• kostas230
In summary, the conversation discusses the proof of a theorem stating that a Hausdorff paracompact space is regular. The conversation focuses on the concept of a locally finite refinement and whether it covers the entire space or just a closed subset. The conclusion is that the refinement covers the entire space and therefore also covers the closed subset.
kostas230
I seem to have stuck an obvious(?) detail in the proof of this theorem. We first show that a Hausdorff paracompact space is regular. Let $X$ be a Hausdorff paracompact space, $K$ be a closed subset of $X$, and $x\in X-K$. Since $X$ is Hausdorff there exists an open cover $\{ V_y: \; y\in K \}$ such that $y\in V_y$and $x \in X-\overline{V}_y$. Let $B$ be a locally finite refinement of the collection $\{ V_y: \; y\in K \}\cup\{X-K\}$ and $U = \cup \{W\in B: \; K\cap W\neq \emptyset\}$. What I cannot understand is why $U$ contains $K$.

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Re-read the definition of ## U ## - is ## K ## one of those sets ## W \in B##?

I think get it now. Willard states that a topological space $X$ is paracomact iff any cover of $X$ has an open locally finite refinement. It does not necessarilly implies that it covers $X$. Munkres however states that this refinement does cover $X$, so $U$ should cover $K$.

Yeah forget what I said, that was silly. The point is that any ## y\in K ## is in some ## W\in B ## since ## B ## is a cover. Obviously for that ##W, y\in W\cap K## so ##W\cap K \neq \emptyset##, hence ##y \in U##

## What is the definition of paracompactness?

Paracompactness is a topological property that states that every open cover of a space can be refined to a locally finite open cover, meaning that every point in the space has a neighborhood that intersects only finitely many open sets in the cover.

## What is the definition of normality in topology?

Normality is a property of a topological space where every pair of disjoint closed sets can be separated by disjoint open sets. In other words, given two disjoint closed sets A and B, there exists open sets U and V, where A is contained in U, B is contained in V, and U and V have no points in common.

## Why is paracompactness important in the study of normality?

Paracompactness is important in the study of normality because it is a necessary condition for a space to be normal. This means that if a space is not paracompact, it cannot be normal.

## What is the relationship between paracompactness and normality?

Paracompactness is a stronger property than normality. This means that every paracompact space is also normal, but not every normal space is paracompact.

## What are some examples of spaces that are both paracompact and normal?

Some examples of spaces that are both paracompact and normal include Hausdorff spaces, metric spaces, and compact spaces. These types of spaces have additional properties that guarantee both paracompactness and normality.

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