Closure of Subset in Subspace/Corollary 1

Corollary to Closure of Subset in Subspace

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 $K \subseteq S$.

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:

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

Proof

\(\ds \map {\cl_H} {K \cap H}\)

\(=\)

\(\ds \map {\cl_T} {K \cap H} \cap H\)

Closure of Subset in Subspace

\(\ds \)

\(\subseteq\)

\(\ds \map {\cl_T} K \cap H\)

Topological Closure of Subset is Subset of Topological Closure

$\blacksquare$