Subspace of Subspace is Subspace

Theorem

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

Let $H \subseteq S$ and $\tau_H$ be the subspace topology on $H$.

Let $K\subseteq H$.

Then the subspace topology on $K$ induced by $\tau$ equals the subspace topology on $K$ induced by $\tau_H$.

Proof

Let $\tau_K$ be the subspace topology on $K$ induced by $\tau$.

Let $\tau’_K$ be the subspace topology on $K$ induced by $\tau_H$.

Then

\(\ds V \in \tau’_K\)

\(\leadstoandfrom\)

\(\ds \exists U’ \in \tau_H : V = U’ \cap K\)

Definition of Subspace Topology $\tau’_K$

\(\ds \)

\(\leadstoandfrom\)

\(\ds \exists U \in \tau : V = \paren {U \cap H} \cap K\)

Definition of Subspace Topology $\tau_H$

\(\ds \)

\(\leadstoandfrom\)

\(\ds \exists U \in \tau : V = U \cap \paren {H \cap K}\)

Intersection is Associative

\(\ds \)

\(\leadstoandfrom\)

\(\ds \exists U \in \tau : V = U \cap K\)

Intersection with Subset is Subset

\(\ds \)

\(\leadstoandfrom\)

\(\ds V \in \tau_K\)

Definition of Subspace Topology $\tau_K$