Closure of Subset in Subspace

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Theorem

Let $T = \struct{S, \tau}$ be a topological space.

Let $H$ be a subset of $S$.

Let $T_H = \struct{H, \tau_H}$ be the topological subspace on $H$.

Let $A$ be a subset of $H$.


Then:

$\map {\operatorname{cl}_H} A = H \cap \map {\operatorname{cl}} A$

where

$\map {\operatorname{cl}_H} A$ denotes the closure of $A$ in $T_H$
$\map {\operatorname{cl}} A$ denotes the closure of $A$ in $T$

Proof

\(\displaystyle \map {\operatorname{cl}_H} A\) \(=\) \(\displaystyle \bigcap \set{K \subseteq H: A \subseteq K, K \text{ is closed in } T_H}\) Definition of closure of subset
\(\displaystyle \) \(=\) \(\displaystyle \bigcap \set{N \cap H: A \subseteq N, N \text{ is closed in } T}\) Closed Set in Topological Subspace
\(\displaystyle \) \(=\) \(\displaystyle H \cap \bigcap \set{N: A \subseteq N, N \text{ is closed in } T}\) Intersection Distributes over Intersection of Family of Sets
\(\displaystyle \) \(=\) \(\displaystyle H \cap \map {\operatorname{cl} } A\) Definition of closure of subset

$\blacksquare$