every second countable space is separable

Theorem 1.

[1]Every second countable space is separable.

Proof.

Let $X$ be a second countable space and let $\\cal B$ be a countable base.For every non-empty set $B$ in $\\cal B$ , choose a point $x_{B}\\in B$ . The set $A$ of all such points $x_{B}$ is clearly countable and it’s also densesince any open set intersects it and thus the whole space is the closure of $A$ .That is, $A$ is a countably dense subset of $X$ . Therefore, $X$ is separable.∎

References

1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.

单词	EverySecondCountableSpaceIsSeparable
释义	every second countable space is separable Theorem 1. [1]Every second countable space is separable. Proof. Let $X$ be a second countable space and let $\\cal B$ be a countable base.For every non-empty set $B$ in $\\cal B$ , choose a point $x_{B}\\in B$ . The set $A$ of all such points $x_{B}$ is clearly countable and it’s also densesince any open set intersects it and thus the whole space is the closure of $A$ .That is, $A$ is a countably dense subset of $X$ . Therefore, $X$ is separable.∎ References 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
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