# Intersection of Interiors contains Interior of Intersection

## Theorem

Let $T$ be a topological space.

Let $\mathbb H$ be a set of subsets of $T$.

That is, let $\mathbb H \subseteq \powerset T$ where $\powerset T$ is the power set of $T$.

Then the interior of the intersection of $\mathbb H$ is a subset of the intersection of the interiors of the elements of $\mathbb H$.

$\displaystyle \paren {\bigcap_{H \mathop \in \mathbb H} H}^\circ \subseteq \bigcap_{H \mathop \in \mathbb H} H^\circ$

## Proof

In the following, $H^-$ denotes the closure of the set $H$.

 $\displaystyle \paren {\bigcap_{H \mathop \in \mathbb H} H}^\circ$ $=$ $\displaystyle T \setminus \paren {T \setminus \bigcap_{H \mathop \in \mathbb H} H}^-$ Complement of Interior equals Closure of Complement $\displaystyle$ $=$ $\displaystyle T \setminus \paren {\paren {\bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H} }^-}$ De Morgan's Laws: Difference with Intersection

At this point we note that:

$(1): \quad \displaystyle \paren {\bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H} }^- \supseteq \bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H}^-$

Then we note that:

$\displaystyle T \setminus \paren {\paren {\bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H} }^-} \subseteq T \setminus \paren {\bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H}^-}$

from $(1)$ and Set Complement inverts Subsets.

Then we continue:

 $\displaystyle T \setminus \paren {\bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H}^-}$ $=$ $\displaystyle T \setminus \paren {\bigcup_{H \mathop \in \mathbb H} T \setminus H^\circ}$ Complement of Interior equals Closure of Complement $\displaystyle$ $=$ $\displaystyle T \setminus \paren {T \setminus \paren {\bigcap_{H \mathop \in \mathbb H} H^\circ} }$ De Morgan's Laws: Difference with Intersection $\displaystyle$ $=$ $\displaystyle \bigcap_{H \mathop \in \mathbb H} H^\circ$ Relative Complement of Relative Complement

$\blacksquare$