Let $X$ be a topological space and $P$ its infinite subbase. Prove that $ w(X) \leq \operatorname{card}(P) $

Asked Apr 20 '19 at 14:04

Active Apr 20 '19 at 14:33

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Let $X$ be a topological space and $P$ its infinite subbase. Prove that $ w(X) \leq \operatorname{card}(P) $, where $w(X)$ is the weight of space X i.e. the minimal cardinality of all bases of $X$.

What i thought of: I would take $B$- a set of all finite intersections of elements of $P$, $B$ is a base of $X$ so $w(X) \leq \operatorname{card}(B)$.

Now it would be great if $\operatorname{card}(B) \leq \operatorname{card}(P)$, but I don't know how to prove it if it's valid. Maybe some set-theory theorem would help.

edited Apr 20 '19 at 14:33

Asaf Karagila

405,794

asked Apr 20 '19 at 14:04

user15269

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Hint: The cardinality of the set of all finite subsets of an infinite set. Notice there is a natural surjection f: the set of finite subsets of a subbase to the set of finite intersections of the elements of the subbase – YuiTo Cheng Apr 20 '19 at 14:18

Let $X$ be a topological space and $P$ its infinite subbase. Prove that $ w(X) \leq \operatorname{card}(P) $

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