Sigma-Locally Finite Cover and Countable Locally Finite Cover have Common Locally Finite Refinement

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Theorem

Let $T = \struct {X, \tau}$ be a topological space.


Let:

$\SS = \ds \bigcup_{n \mathop = 0}^\infty \SS_n$ be a $\sigma$-locally finite cover of $X$

where each $\SS_n$ is locally finite for all $n \in \N$.

Let $\CC = \set {C_n : n \in \N}$ be a countable locally finite cover of $X$.


Then:

there exists a common locally finite refinement $\AA$ of both $\SS$ and $\CC$


Proof

Let:

$\AA = \set {C_n \cap S : n \in \N, S \in \SS_n}$


$\AA$ is a Cover of $X$

Let $x \in X$.


By definition of a cover:

$\exists n \in \N : x \in C_n$

and

$\exists S \in \SS : x \in S$

By definition of set union:

$\exists n \in \N : S \in \SS_n$

By definition of set intersection:

$x \in C_n \cap S$

By definition of $\AA$:

$C_n \cap S \in \AA$


It follows that $\AA$ is a cover by definition.

$\Box$


$\AA$ is a Refinement of $\SS$ and $\CC$

Let $A \in \AA$.

By definition of $\AA$:

$\exists n \in \N, S \in SS_n : A = C_n \cap S$

From Intersection is Subset:

$A \subseteq C_n$ and $A \subseteq S$

Hence $\AA$ is a refinement of both $\SS$ and $\CC$ by definition.

$\Box$


$\AA$ is Locally Finite

Let $x \in X$.


By definition of locally finite:

$\exists U \in \tau : x \in U : \set {C_n \in \CC : C_n \cap U \ne \O}$ is finite.


From Subset of Naturals is Finite iff Bounded:

$N = \set {n \in \N : C_n \cap U \ne \O}$ is bounded.

Hence $N$ has a greatest element $m$.


By definition of locally finite:

$\forall n \le m : \exists V_n \in \tau : x \in V_n : \set {S \in \SS_n : S \cap V_n \ne \O}$ is finite.


Let:

$W = U \cap \ds \bigcap_{n \mathop \le m} V_n$

By Open Set Axiom $\paren {\text O 2 }$: Pairwise Intersection of Open Sets:

$W \in \tau$


Let $A \in \AA : A \cap W \ne \O$.

Hence there exists $n \in \N, S \in \SS_n:$

$A = C_n \cap S$

From Subsets of Disjoint Sets are Disjoint:

$C_n \cap U \ne \O$

Hence $n \le m$.

From Subsets of Disjoint Sets are Disjoint:

$S \cap V_n \ne \O$


We have:

\(\ds \set {A \in \AA : A \cap W \ne \O}\) \(\subseteq\) \(\ds \set {C_n \cap S : n \le m, S \in \SS_n, S \cap V_n \ne \O}\)
\(\ds \) \(=\) \(\ds \bigcup_{n \mathop \le m} \set {C_n \cap S : S \in \SS_n, S \cap V_n \ne \O}\)


From Finite Union of Finite Sets is Finite:

$\ds \bigcup_{n \mathop \le m} \set {C_n \cap S : S \in \SS_n, S \cap V_n \ne \O}$ is finite.

From Subset of Finite Set is Finite:

$\set {A \in \AA : A \cap W \ne \O}$ is finite.


Hence $\AA$ is locally finite by definition.

$\blacksquare$

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