Characterization of Paracompactness in T3 Space/Lemma 10
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Theorem
Let $T = \struct{X, \tau}$ be a topological space.
Let $\UU$ be an open cover of $T$.
Let $\AA = \ds \bigcup_{n \in \N} \AA_n$ be a $\sigma$-discrete refinement of $\UU$:
- $\forall n \in \N : \AA_n$ is a discrete set of subsets
For each $n \in \N$, let $V_n$ be an open neighborhood of the diagonal $\Delta_X$ of $X \times X$ in the product space $\struct {X \times X, \tau_{X \times X} }$:
- $\forall x \in X : \card {\set{A \in \AA_n : \map {V_n} x \cap V_n \sqbrk A \ne \O}} \le 1$
For each $A \in \AA$ let:
- $U_A \in U : A \subseteq U_A$
For each $n \in \N$, let:
- $\WW_n = \set{U_A \cap V_n \sqbrk A : A \in \AA_n}$
Let:
- $\WW = \ds \bigcup_{n \in \N} \WW_n$
Then:
- $\WW$ is an open $\sigma$-discrete refinement of $\UU$
Proof
$\WW$ is Set of Open Sets
Let:
- $W \in \WW$
By definition of $\WW$:
- $\exists n \in \N, A \in \AA : W = U_A \cap V_n \sqbrk A$
We have by hypothesis:
- $U_A \in \tau$
From Image of Subset under Open Neighborhood of Diagonal is Open Neighborhood of Subset:
- $V_n \sqbrk A \in \tau$
By Open Set Axiom $\paren {\text O 2 }$: Pairwise Intersection of Open Sets:
- $W \in \tau$
Since $W$ was arbitrary, it follows that:
$\Box$
$\WW$ is a Cover of $X$
Let:
- $x \in X$
By definition of cover of set:
- $\exists A \in \AA : x \in A$
Let:
- $n = \min \set{m \in \N : A \in \AA_m}$
By definition of neighborhood:
- $\Delta_X \subseteq V_n$
Hence:
- $\forall a \in A : \tuple{a, a} \in V_n$
By definition of image:
- $\forall a \in A : a \in V_n \sqbrk A$
By definition of subset:
- $A \subseteq V_n \sqbrk A$
We have by hypothesis:
- $A \subseteq U_A$
From Set is Subset of Intersection of Supersets:
- $A \subseteq U_A \cap V_n \sqbrk A$
By definition of subset:
- $x \in U_A \cap V_n \sqbrk A$
By definition of $\WW_n$:
- $U_A \cap V_n \sqbrk A \in \WW_n \subseteq \WW$
Hence:
- $\exists W \in \WW : x \in W$
Since $x$ was arbitrary, it follows that $\WW$ is a cover of $X$.
$\Box$
$\WW$ is a Refinement of $\UU$
Let:
- $W \in \WW$
By definition of $\WW$:
- $\exists n \in \N, A \in \AA : W = U_A \cap V_n \sqbrk A$
From Intersection is Subset:
- $W \subseteq U_A$
Hence:
- $\exists U \in \UU : W \subseteq U$
Since $W$ was arbitrary, it follows that $\WW$ is a refinement of $\UU$ by definition.
$\Box$
$\WW_k$ is Discrete
Let $n \in \N$.
Let $x \in X$.
We have by hypothesis:
- $\card {\set{A \in \AA_n : \map {V_n} x \cap V_n \sqbrk A \ne \O}} \le 1$
From Subsets of Disjoint Sets are Disjoint:
- $\forall A \in A_n : \map {V_n} x \cap V_n \sqbrk A \cap U_A \ne \O \leadsto \map {V_n} x \cap V_n \sqbrk A \ne \O$
Hence:
- $\set{A \in \AA_n : \map {V_n} x \cap V_n \sqbrk A \cap U_A \ne \O} \subseteq \set{A \in \AA_n : \map {V_n} x \cap V_n \sqbrk A \ne \O}$
From Cardinality of Subset of Finite Set:
- $\card{\set{A \in \AA_n : \map {V_n} x \cap V_n \sqbrk A \cap U_A \ne \O}} \le 1$
The mapping $f: \set{A \in \AA_n : \map {V_n} x \cap V_n \sqbrk A \cap U_A \ne \O} \to \set{W \in \WW_n : \map {V_n} x \cap W \ne \O}$ defined by:
- $\map f A = V_n \sqbrk A \cap U_A$
is surjective.
From Cardinality of Surjection:
- $\card {\set{W \in \WW_n : \map {V_n} x \cap W \ne \O}} \le \card {\set{A \in \AA_n : \map {V_n} x \cap V_n \sqbrk A \cap U_A \ne \O}}$
Hence:
- $\card {\set{W \in \WW_n : \map {V_n} x \cap W \ne \O}} \le 1$
From Image of Point under Open Neighborhood of Diagonal is Open Neighborhood of Point:
- $\map {V_n} x \in \tau$ is an open neighborhood of $x$
Since $x$ was arbitrary, it follows that:
- $\forall x \in X : \map {V_n} x \in \tau : x \in \map {V_n} x : \card {\set{W \in \WW_n : \map {V_n} x \cap W \ne \O}} \le 1$
It follows that $\WW_n$ is a discrete set of subsets by definition.
Since $n$ was arbitrary, it follows that:
- $\forall n \in \N : \WW_n$ is a discrete set of subsets
$\Box$
It follows that $\WW$ is an open $\sigma$-discrete refinement of $\UU$ by definition.
$\blacksquare$