Subspace of Subspace is Subspace

Theorem

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

Let $H \subseteq S$ be a non-empty subset of $S$ and $\tau_H$ be the subspace topology on $H$.

Let $K \subseteq S$ be a non-empty subset of $S$.

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

This article, or a section of it, needs explaining.
In particular: Why is it necessary to state that $H$ and $K$ are non-empty?
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.

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 $\tau_K$

$\blacksquare$

Sources

2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: New Spaces From Old: Subspaces