Interior of Subset
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Theorem
Let $\left({S, \tau}\right)$ be a topological space.
Let $X$ and $Y$ be subsets of $S$, and suppose that $X \subseteq Y$.
Then:
- $X^\circ \subseteq Y^\circ$
where $X^\circ$ denotes the interior of $X$.
Proof
By definition of interior, $X^\circ$ is open in $\tau$, and:
- $Y^\circ \subseteq Y$
Hence, by Subset Relation is Transitive:
- $X^\circ \subseteq Y$
is an open set contained in $Y$.
The result follows by Set Interior is Largest Open Set.
$\blacksquare$