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**I-Think** Question

Let$\displaystyle E^o$ be the set of all interior points of a set $\displaystyle E$.

Prove $\displaystyle E^o$ is always open

Proof

Consider $\displaystyle p\in{E^o}$. $\displaystyle \exists$ a neighborhood $\displaystyle N$ of $\displaystyle p$ such that $\displaystyle N\subset{E}$

This $\displaystyle N$ is an open set, so $\displaystyle \forall{q\in{N}}, \exists$ a neighborhood $\displaystyle M$ of $\displaystyle q$ so $\displaystyle M\subset{N\subset{E}}$

So $\displaystyle N$ consists of interior points of $\displaystyle E$, so $\displaystyle E^o$ is open.

QED