Finite Intersection of Regular Open Sets is Regular Open

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Theorem

Let $T$ be a topological space.

Let $n \in \N$.


Suppose that:

$\forall i \in \left[{1 \,.\,.\, n}\right]: H_i \subseteq T$

where all the $H_i$ are regular open in $T$, i.e.:

$\forall i \in \left[{1 \,.\,.\, n}\right]: H_i = H_i^{- \circ}$


Then $\displaystyle \bigcap_{i \mathop = 1}^n H_i$ is regular open in $T$.

That is:

$\displaystyle \bigcap_{i \mathop = 1}^n H_i = \left({\bigcap_{i \mathop = 1}^n H_i}\right)^{- \circ}$


Proof

\(\displaystyle \left({\bigcap_{i \mathop = 1}^n H_i}\right)^{- \circ}\) \(=\) \(\displaystyle \left({T \setminus \left({T \setminus \bigcap_{i \mathop = 1}^n H_i}\right)^\circ}\right)^\circ\) Complement of Interior equals Closure of Complement
\(\displaystyle \) \(=\) \(\displaystyle \left({T \setminus \left({\bigcup_{i \mathop = 1}^n \left({T \setminus H_i}\right)}\right)^\circ}\right)^\circ\) De Morgan's Laws: Difference with Intersection
\(\displaystyle \) \(=\) \(\displaystyle \left({\left({T \setminus \bigcup_{i \mathop = 1}^n \left({T \setminus H_i}\right)}\right)^-}\right)^\circ\) Complement of Interior equals Closure of Complement
\(\displaystyle \) \(=\) \(\displaystyle \left({\bigcap_{i \mathop = 1}^n \left({T \setminus \left({T \setminus H_i}\right)}\right)^-}\right)^\circ\) De Morgan's Laws: Difference with Intersection
\(\displaystyle \) \(=\) \(\displaystyle \left({\bigcap_{i \mathop = 1}^n H_i^-}\right)^\circ\) Relative Complement of Relative Complement
\(\displaystyle \) \(=\) \(\displaystyle \bigcap_{i \mathop = 1}^n H_i^{- \circ}\) Interior of Finite Intersection equals Intersection of Interiors
\(\displaystyle \) \(=\) \(\displaystyle \bigcap_{i \mathop = 1}^n H_i\) as all $H_i$ are regular open in $T$

$\blacksquare$


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