Cardinality of Image of Mapping of Intersections is not greater than Weight of Space
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Theorem
Let $T = \struct {X, \tau}$ be a topological space.
Let $f: X \to \tau$ be a mapping such that:
- $\forall x \in X: \paren {x \in \map f x \land \forall U \in \tau: x \in U \implies \map f x \subseteq U}$
Then the cardinality of the image of $f$ is no greater than the weight of $T$:
- $\card {\Img f} \le \map w T$
Proof
By definition of weight, there exists a basis $\BB$ of $T$ such that:
- $\card \BB = \map w T$
By Image of Mapping of Intersections is Smallest Basis:
- $\Img f \subseteq \BB$
Thus by Subset implies Cardinal Inequality:
- $\card {\Img f} \le \card \BB = \map w T$
$\blacksquare$
Sources
- Mizar article TOPGEN_2:17