Equivalence of Definitions of Weight of Topological Space

Theorem
Let $T$ be a topological space.

Let $\mathbb B$ be the set of all bases of $T$.

The following definitions of the weight of $T$ are equivalent:

Proof
By Cardinal Class is Subset of Ordinal Class, the set:
 * $M = \left\{{\left\vert {\mathcal B} \right\vert: \mathcal B \in \mathbb B}\right\}$

is a subset of the class of ordinals.

By Ordinal Class is Strongly Well-Ordered by Subset:
 * $M$ is well ordered by the $\subseteq$ relation.

By Ordinal Class is Strongly Well-Ordered by Subset there exists a smallest element $m_0 \in M$:
 * $\forall m \in M: m_0 \subseteq m$

Hence by Smallest Element is Minimal there exists a basis $\mathcal B_0$ of $T$ which has minimal cardinality:
 * $m_0 = w_2 \left({T}\right)$.

Let:
 * $\displaystyle w_1 \left({T}\right) = \bigcap_{\mathcal B \mathop \in \mathbb B} \left\vert {\mathcal B} \right\vert$

By Intersection is Subset:


 * $\displaystyle w_1 \left({T}\right) = \bigcap M \subseteq m_0$

But by Intersection is Largest Subset:


 * $\displaystyle \mathfrak m_0 \subseteq \bigcap M$

By definition of set equality:
 * $w_1\left({T}\right) = w_2\left({T}\right)$

and hence the result.