Definition:Weight of Topological Space

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Let $T$ be a topological space.

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

Definition 1

The weight of $T$ is defined as:

$\ds \map w T := \bigcap_{\BB \mathop \in \mathbb B} \card \BB$

where $\card \BB$ denotes the cardinality of $\BB$.

Definition 2

The weight of $T$ is the smallest cardinality of the elements of $\mathbb B$:

$\map w T := \min \set {\card \BB: \BB \in \mathbb B}$

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