Subspace of Subspace is Subspace
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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 $\tau_K$ |