# Interior of Subset

## 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.

$\blacksquare$