The Interior is the Largest Open Subset

Let $(X,\mathcal{T})$ be a topological space, and let $A\subseteq X$. Let $\operatorname{int}_X(A)$ denote the interior of $A$ in $X$.

Then the following hold.

1. $\operatorname{int}_X(A)\subseteq A$.

2. $\operatorname{int}_X(A)\in\mathcal{T}$.

3. If $U\in\mathcal{T}$ and $U\subseteq A$, then $U\subseteq\operatorname{int}_X(A)$.

4. $A\in\mathcal{T}$ if and only if $A=\operatorname{int}_X(A)$.

5. If $B\subseteq X$ and $A\subseteq B$, then $\operatorname{int}_X(A)\subseteq\operatorname{int}_X(B)$.