Interior of Subset in Subspace

Theorem

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

Let $T_H = \struct {H, \tau_H}$ be a subspace of $T$ where $H \subseteq S$.

Let $A \subseteq H$ be an arbitrary subset of $H$.


Then:

$\map {\mathrm {Int}_H} A \supseteq H \cap \map {\mathrm {Int} } A$

where:

$\map {\mathrm {Int}_H} A$ denotes the interior of $A$ in $T_H$
$\map {\mathrm {Int} } A$ denotes the interior of $A$ in $T$.


Proof


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Sources

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