Density not greater than Weight

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Then
 * $d \left({T}\right) \leq w \left({T}\right)$

where
 * $d \left({T}\right)$ denotes the density of $T$,
 * $w \left({T}\right)$ denotes the weight of $T$.

Proof
By definition of weight there exists a basis $\mathcal B$ of $T$:
 * $w \left({T}\right) = \left\vert{\mathcal B}\right\vert$

where $\left\vert{\mathcal B}\right\vert$ denotes the cardinality of $\mathcal B$.

By Axiom of Choice define a mapping $f: \left\{{U \in \mathcal B: U \ne \varnothing}\right\} \to S$:
 * $\forall U \in \mathcal B: U \ne \varnothing \implies f \left({U}\right) \in U$

We will prove that
 * $\forall U \in \tau: U \ne \varnothing \implies U \cap \operatorname{Im} \left({f}\right) \ne \varnothing$