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Closure of a Set in a Metric Space is all its Sequential Limits

Last updated Nov 1, 2022

# Statement

Suppose (X,d)(X, d) is a Metric Space

Metric Space

Definition A is a MM equipped with a d:M×MR0d: M \times M \to \mathbb{R}_{\geq0}. Remarks A is naturally endowed...

11/7/2022

. Suppose SXS \subset X Then clS=x:(xn)S s.t. xnx\text{cl} S = {x : \exists (x_n) \subset S \text{ s.t. } x_{n} \to x}

# Proof

Metric Spaces are First Countable and Hausdorff and Closure of a Set in a First Countable Space is all its Sequential Limits

Closure of a Set in a First Countable Space is all its Sequential Limits

Statement Suppose (X,τ)(X, \tau) is a . Suppose SXS \subset X Then $$\text{cl} S = \{x : \exists (xn) \subset...

11/7/2022

. \blacksquare