Characterization of Paracompactness in T3 Space/Statement 3 implies Statement 1
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Theorem
Let $T = \struct{X, \tau}$ be a topological space.
If every open cover of $T$ have a closed locally finite refinement then:
- $T$ is paracompact.
Proof
Let every open cover of $T$ have a closed locally finite refinement.
Let $\UU$ be an open cover of $T$.
Let $\VV$ be a closed locally finite refinement of $\UU$, which exists by assumption.
Let $\WW = \set{W \in \tau : \set{V \in \VV : V \cap W \ne \O} \text{ is finite}}$.
By definition of locally finite:
- $\forall x \in X: \exists W \in \tau: x \in W$ and $\set{V \in \VV : V \cap W \ne \O}$ is finite.
Hence $\WW$ is an open cover of $T$, by definition.
Let $\AA$ be a closed locally finite refinement of $\WW$, which exists by assumption.
Lemma 4
- $\forall A \in \AA : \set{V \in \VV : V \cap A \ne \O}$ is finite
$\Box$
For each $V \in \VV$, let:
- $V^* = X \setminus \ds \bigcup \set{A \in \AA : A \cap V = \O}$
Lemma 5
- $\forall V \in \VV: V \subseteq V^*$
$\Box$
Let $\VV^* = \set{V^* : V \in \VV}$.
Lemma 6
- $\VV^*$ is an open locally finite cover of $T$
$\Box$
From Lemma 5 and Lemma 6 it follows that $\VV$ is a refinement of $\VV^*$ by definition.
By definition of refinement:
- $\forall V \in \VV : \exists U \in \UU : V \subseteq U$
For each $V \in \VV$, let:
- $U_V \in \UU : V \subseteq U_V$
Let:
- $\UU^* = \set{V^* \cap U_V : V \in \VV}$
Lemma 7
- $\UU^*$ is an open locally finite refinement of $\UU$
$\Box$
Since $\UU$ was arbitrary, it follows that $T$ is paracompact by definition.
$\blacksquare$
Sources
- 1970: Stephen Willard: General Topology: Chapter $6$: Compactness: $\S20$: Paracompactness: Theorem $20.7$