Characterization of Paracompactness in T3 Space/Lemma 6
Theorem
Let $T = \struct {X, \tau}$ be a topological space.
Let $\UU$ be an open cover of $T$.
Let $\VV$ be a closed locally finite refinement of $\UU$.
For all $x \in X$, let:
- $W_x \in \tau: x \in W_x$ and $\set {V \in \VV : V \cap W \ne \O}$ is finite
Let $\WW = \set {W_x : x \in X}$ be an open cover of $T$.
Let $\AA$ be a closed locally finite refinement of $\WW$.
For each $V \in \VV$, let:
- $V^* = X \setminus \bigcup \set {A \in \AA | A \cap V = \O}$
Let $\VV^* = \set {V^* : V \in \VV}$.
Then:
- $\VV^*$ is an open locally finite cover of $T$
Proof
Lemma 4
- $\forall A \in \AA : \set{V \in \VV : V \cap A \ne \O}$ is finite
$\Box$
Lemma 11
- $\forall A \in \AA, V^* \in \VV^* : A \cap V^* \ne \O \implies A \cap V \ne \O$
$\Box$
$\VV^*$ is a Set of Open Subsets
Let $V^* \in \VV^*$ for some $V \in \VV$.
Let $\AA_V = \set {A \in \AA | A \cap V = \O}$.
By definition of subset:
- $\AA_V \subseteq \AA$
From Subset of Locally Finite Set of Subsets is Locally Finite:
- $\AA_V$ is closed locally finite
From Union of Closed Locally Finite Set of Subsets is Closed:
- $\bigcup \set{A \in \AA | A \cap V = \O}$ is closed in $T$
By definition of closed set:
- $V^* = X \setminus \ds \bigcup \set{A \in \AA | A \cap V = \O} \in \tau$
Since $V^*$ was arbitrary, it follows that:
- $V^* \in \VV^* : V^* \in \tau$
$\Box$
$\VV^*$ is a Cover
Let $x \in X$.
By definition of a cover:
- $\exists V \in \VV : x \in V$
From Lemma $4$:
- $V \subseteq V^*$
By definition of subset:
- $x \in V^*$
Since $x$ was arbitrary, it follows that $\VV^*$ is a cover by definition.
$\Box$
$\VV^*$ is Locally Finite
Let $x \in X$.
By definition of locally finite:
- $\exists U \in \tau : x \in U : \set{A \in \AA : A \cap U \ne \O}$ is finite.
Let $\set{A \in \AA : A \cap U \ne \O} = \set{A_1, A_2, \ldots, A_n}$ for some $n \in \N$.
From Subset of Cover is Cover of Subset:
- $U \subseteq \ds \bigcup \set{A_1, A_2, \ldots, A_n}$
Let:
- $V^* \in \VV^* : V^* \cap U \ne \O$
We have:
\(\ds \O\) | \(\ne\) | \(\ds V^* \cap \bigcup_{i = 1}^n A_i\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \bigcup_{i = 1}^n V^* \cap A_i\) | Intersection Distributes over Union |
Hence:
- $\exists i \in \set{1, 2, \ldots, n} : V^* \cap A_i \ne \O$
From Lemma $11$:
- $V \cap A_i \ne \O$
Hence:
- $\set {V^* \in \VV^* : V^* \cap U} \subseteq \set {V^* \in \VV^* : \exists 1 \le i \le n : V \cap A_i \ne \O}$
For each $1 \le i \le n$:
- $\exists W_i \in \WW : A_i \subseteq W_i$
By definition of $\WW$:
- $\forall 1 \le i \le n : \set {V \in \VV : V \cap W_i \ne \O}$ is finite
Hence:
- $\forall 1 \le i \le n : \set {V \in \VV : V \cap A_i \ne \O}$ is finite
From Union of Finite Sets is Finite:
- $\set {V^* \in \VV^* : \exists 1 \le i \le n : V \cap A_i \ne \O}$ is finite
From Subset of Finite Set is Finite:
- $\set {V^* \in \VV^* : V^* \cap U}$ is finite
Since $x$ was arbitrary, it follows that $\VV^*$ is locally finite.
$\blacksquare$