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A Set is Closed in a First Countable Space iff it contains all its Sequential Limits

Last updated Nov 1, 2022

# Statement

Suppose $(X, \tau)$ is a First Countable Space

. Then

K \subset X

is Closed If and Only If

K \supset {x : \exists (x_n) \subset K \text{ s.t. } x_{n} \to x}

That is,

K

contains all its sequential limits.

# Proof