Interior is the largest open subset

Statement

Let $X$ be a topological space and let $S \subseteq X$. Then $\operatorname{int} S \subseteq S$, the set $\operatorname{int} S$ is open, and each open set $U$ of $X$ with $U \subseteq S$ satisfies $U \subseteq \operatorname{int} S$.

Remarks

Here $\operatorname{int} S$ is the interior of $S$. The openness is the statement interior is open. A consequence: $S$ is open if and only if $\operatorname{int} S = S$. If $S \subseteq T$, then $\operatorname{int} S \subseteq \operatorname{int} T$.

Explanations

L1 human

Among all open sets inside $S$, the interior is the biggest one. Each open set inside $S$ is already inside the interior.