Open Cover with Closed Locally Finite Refinement is Even Cover/Lemma 3

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Theorem

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

Let $\UU$ be an open cover of $T$.

Let $\AA$ be a closed locally finite refinement of $\UU$.

For each $A \in \AA$, let $U_A \in \UU$ such that $A \subseteq U_A$.

For each $A \in \AA$, let:

$V_A = \paren {U_A \times U_A} \cup \paren {\paren {X \setminus A} \times \paren {X \setminus A} }$

For each $x \in X, A \in \AA$, let:

$\map {V_A} x = \set {y \in X : \tuple {x, y} \in V_A}$

where:

$V_A$ is seen as a relation on $X \times X$

$\map {V_A} x$ denotes the image of $x$ under $V_A$.

Then:

$\forall A \in \AA, x \in A : \map {V_A} x = U_A$

Proof

We have:

\(\ds \forall A \in \AA, x \in A: \, \)

\(\ds \map {V_A} x\)

\(=\)

\(\ds \set {y \in X : \tuple{x, y} \in V_A}\)

Definition of $\map {V_A} x$

\(\ds \)

\(=\)

\(\ds \set {y \in X : \tuple {x, y} \in \paren {U_A \times U_A} \cup \paren {\paren {X \setminus A} \times \paren {X \setminus A} } }\)

Definition of $V_A$

\(\ds \)

\(=\)

\(\ds \set {y \in X : \tuple {x, y} \in U_A \times U_A \text{ or } \tuple {x, y} \in \paren {X \setminus A} \times \paren {X \setminus A} }\)

Definition of Set Union

\(\ds \)

\(=\)

\(\ds \set {y \in X : \tuple {x, y} \in U_A \times U_A}\)

as $x \in A$

\(\ds \)

\(=\)

\(\ds \set {y \in X : y \in U_A}\)

Definition of Cartesian Product

\(\ds \)

\(=\)

\(\ds U_A\)

Definition of Set

$\blacksquare$