Open sets and sets of interior points

**I-Think** · August 9th 2011, 03:56 AM

Need verification of a proof please

Question
Let $E^o$ be the set of all interior points of a set $E$ .
Prove $E^o$ is always open

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

This $N$ is an open set, so $\forall{q\in{N}}, \exists$ a neighborhood $M$ of $q$ so $M\subset{N\subset{E}}$
So $N$ consists of interior points of $E$ , so $E^o$ is open.
QED

Proof correct? Any alternatives or ways to make it cleaner/more elegant?

**Plato** · August 9th 2011, 04:10 AM

Originally Posted by I-Think

Question
Let $E^o$ be the set of all interior points of a set $E$ .
Prove $E^o$ is always open
Proof
Consider $p\in{E^o}$ . $\exists$ a neighborhood $N$ of $p$ such that $N\subset{E}$
This $N$ is an open set, so $\forall{q\in{N}}, \exists$ a neighborhood $M$ of $q$ so $M\subset{N\subset{E}}$
So $N$ consists of interior points of $E$ , so $E^o$ is open.
QED

It is correct. Basically you are noting that a neighborhood is a neighborhood of each of its points.

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