Existence of Subfamily of Cardinality not greater than Weight of Space and Unions Equal
Theorem
Let $T$ be a topological space.
Let $\mathcal F$ be a set of open sets of $T$.
There exists a subset $\mathcal G \subseteq \mathcal F$ such that:
- $\displaystyle \bigcup \mathcal G = \bigcup \mathcal F$
and:
- $\left\vert{\mathcal G}\right\vert \leq w \left({T}\right)$
where:
- $w \left({T}\right)$ denotes the weight of $T$
- $\left\vert{\mathcal G}\right\vert$ denotes the cardinality of $\mathcal G$.
Proof
By definition of weight of $T$ there exists a basis $\mathcal B$ of $T$ such that:
- $(1): \quad \left\vert{\mathcal B}\right\vert = w \left({T}\right)$
Let:
- $\mathcal B_1 = \left\{{W \in \mathcal B: \exists U \in \mathcal F: W \subseteq U}\right\}$
By definition of subset:
- $\mathcal B_1 \subseteq \mathcal B$
Then by Subset implies Cardinal Inequality:
- $(2): \quad \left\vert{\mathcal B_1}\right\vert \leq \left\vert{\mathcal B}\right\vert$
By definition of set $\mathcal B_1$:
- $\forall W \in \mathcal B_1: \exists U \in \mathcal F: W \subseteq U$
Then by the Axiom of Choice there exists a mapping $f$ from $\mathcal B_1$ into $\mathcal F$ such that
- $(3): \quad \forall U \in \mathcal B_1: U \subseteq f \left({U}\right)$
Let:
- $\mathcal G = \operatorname{Im} \left({f}\right)$
where $\operatorname{Im} \left({f}\right)$ denotes the image of $f$.
- $\mathcal G \subseteq \mathcal F$
Then by Union of Subset of Family is Subset of Union of Family:
- $\displaystyle \bigcup \mathcal G \subseteq \bigcup \mathcal F$
By definition of set equality, to prove $\displaystyle \bigcup \mathcal G = \bigcup \mathcal F$ it is sufficient to show:
- $\displaystyle \bigcup \mathcal F \subseteq \bigcup \mathcal G$
Let $\displaystyle x \in \bigcup \mathcal F$.
By definition of union there exists a set $A$ such that:
- $A \in \mathcal F$ and $x \in A$
Because $A$ is open, then by definition of basis there exists $U \in \mathcal B$ such that:
- $x \in U \subseteq A$
By definition of the set $\mathcal B_1$:
- $U \in \mathcal B_1$
Because $\mathcal G = \operatorname{Im} \left({f}\right)$:
- $f \left({U}\right) \in \mathcal G$
By $(3)$ and Set is Subset of Union:
- $\displaystyle U \subseteq f \left({U}\right) \subseteq \bigcup \mathcal G$
Thus by definition of subset:
- $\displaystyle x \in \bigcup \mathcal G$
This ends the proof of inclusion.
By Cardinality of Image of Mapping not greater than Cardinality of Domain:
- $\left\vert{\operatorname{Im} \left({f}\right)}\right\vert \leq \left\vert{\mathcal B_1}\right\vert$
Thus by $(1)$ and $(2)$:
- $\left\vert{\mathcal G}\right\vert \leq w \left({T}\right)$
$\blacksquare$
Sources
- Mizar article TOPGEN_2:11