Closure of Subset in Subspace

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Theorem

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

Let $H \subseteq S$ be an arbitrary subset of $S$.

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

Let $A \subseteq H$ be an arbitrary subset of $H$.


Then:

$\map {\cl_H} A = H \cap \map \cl A$

where:

$\map {\cl_H} A$ denotes the closure of $A$ in $T_H$
$\map \cl A$ denotes the closure of $A$ in $T$.


Corollary 1

Let $K \subseteq S$.

Let $\map {\cl_T} K$ denote the closure of $K$ in $T$.

Let $\map {\cl_H} {K \cap H}$ denote the closure of $K \cap H$ in $T_H$.


Then:

$\map {\cl_H} {K \cap H} \subseteq \map {\cl_T} K \cap H$


Corollary 2

Let $H$ be closed in $T$.

Then:

$\map {\cl_H} A = \map \cl A$


Proof

\(\ds \map {\cl_H} A\) \(=\) \(\ds \bigcap \set {K \subseteq H: A \subseteq K, K \text{ is closed in } T_H}\) Definition of Closure (Topology)
\(\ds \) \(=\) \(\ds \bigcap \set {N \cap H: A \subseteq N, N \text{ is closed in } T}\) Closed Set in Topological Subspace
\(\ds \) \(=\) \(\ds H \cap \bigcap \set {N: A \subseteq N, N \text{ is closed in } T}\) Set Intersection is Self-Distributive over Family of Sets
\(\ds \) \(=\) \(\ds H \cap \map \cl A\) Definition of Closure (Topology)

$\blacksquare$


Sources

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