Closure of Subset in Subspace

From ProofWiki
Jump to navigation Jump to search

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:

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

where

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


Proof

\(\displaystyle \map {\cl_H} A\) \(=\) \(\displaystyle \bigcap \set {K \subseteq H: A \subseteq K, K \text{ is closed in } T_H}\) Definition of Closure (Topology)
\(\displaystyle \) \(=\) \(\displaystyle \bigcap \set {N \cap H: A \subseteq N, N \text{ is closed in } T}\) 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
\(\displaystyle \) \(=\) \(\displaystyle H \cap \map \cl A\) Definition of Closure (Topology)

$\blacksquare$