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:
 * $M = \set {\card \BB: \BB \in \mathbb B}$

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:
 * $m_0 = \map {w_2} T$.

Let:
 * $\displaystyle \map {w_1} T = \bigcap_{\BB \mathop \in \mathbb B} \card \BB$

By Intersection is Subset:


 * $\displaystyle \map {w_1} T = \bigcap M \subseteq m_0$

But by Intersection is Largest Subset:


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

By definition of set equality:
 * $\map {w_1} T = \map {w_2} T$

and hence the result.