Characterization of Paracompactness in T3 Space/Lemma 20

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Theorem

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


Let every open cover of $T$ be even.


Let $\tau_{X \times X}$ denote the product topology on the cartesian product $X \times X$.


Let $N$ be a neighborhood of the diagonal $\Delta_X$ of $X \times X$ in the product space $\struct {X \times X, \tau_{X \times X} }$.


Then there exists an open neighborhood $W$ of the diagonal $\Delta_X$ in $\struct {X \times X, \tau_{X \times X} }$:

$W$ is symmetric as a relation on $X \times X$, that is, $W = W^{-1}$
the composite relation $W \circ W$ is a subset of $N$, that is, $W \circ W \subseteq N$

Proof

Let:

$\VV = \set{V \in \tau : V \times V \subseteq N}$


From Neighborhood of Diagonal induces Open Cover:

$\VV$ is an open cover of $T$


We have by hypothesis, $\VV$ is even.


From Characterization of Even Cover, there exists an open neighborhood $R$ of the diagonal $\Delta_X$ in $\struct {X \times X, \tau_{X \times X} }$:

$\set{\map R x : x \in S}$ is a refinement of $\VV$

where:

$R$ is seen as a relation on $X \times X$
$\map R x$ denotes the image of $x$ under $R$.


By definition of refinement of cover:

$\forall x \in X : \exists V \in \VV : \map R x \subseteq V$


From Cartesian Product of Subsets:

$\forall x \in X : \exists V \in \VV : \map R x \times \map R x \subseteq V \times V$


From Subset Relation is Transitive:

$\forall x \in X : \map R x \times \map R x \subseteq N$


Let $W = R \cap R^{-1}$, where $R^{-1}$ is the inverse relation of $R$ on $X \times X$.


From Inverse of Open Set in Product Space is Open in Inverse Product Space:

$R^{-1}$ is open in $\struct {X \times X, \tau_{X \times X}}$


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

$W$ is open in $\struct {X \times X, \tau_{X \times X}}$


From Inverse of Reflexive Relation is Reflexive:

$R^{-1}$ is reflexive


From Intersection of Reflexive Relations is Reflexive:

$W$ is reflexive


By definition of reflexive:

$W$ is an open neighborhood of the diagonal $\Delta_X$.


From Intersection of Relation with Inverse is Symmetric Relation:

$W$ is a symmetric relation on $X \times X$


We have:

\(\ds \forall x, y, z \in X: \, \) \(\ds \tuple{y,z} \in \map W x \times \map W x\) \(\leadsto\) \(\ds y, z \in \map W x\) Definition of Cartesian Product
\(\ds \) \(\leadsto\) \(\ds \tuple{x, y}, \tuple{x, z} \in W\) Definition of Image of Element under Relation
\(\ds \) \(\leadsto\) \(\ds \tuple{x, y}, \tuple{x, z} \in R\) Intersection is Subset and Definition of Subset
\(\ds \) \(\leadsto\) \(\ds y, z \in \map R x\) Definition of Image of Element under Relation
\(\ds \) \(\leadsto\) \(\ds \tuple{y, z} \in \map R x \times \map R x\) Definition of Cartesian Product
\(\ds \) \(\leadsto\) \(\ds \tuple{y, z} \in N\) Definition of Subset


By definition of subset:

$\forall x \in X : \map W x \times \map W x \subseteq N$


From Composition of Symmetric Relation with Itself is Union of Products of Images:

$W \circ W = \ds \bigcup_{x \in X} \map W x \times \map W x$


From Union of Subsets is Subset:

$W \circ W \subseteq N$

$\blacksquare$

Sources

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