Open sets and sets of interior points

Need verification of a proof please

Question

Let be the set of all interior points of a set .

Prove is always open

Proof

Consider . a neighborhood of such that

This is an open set, so a neighborhood of so

So consists of interior points of , so is open.

QED

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

Re: Open sets and sets of interior points

Quote:

Originally Posted by

**I-Think** Question

Let

be the set of all interior points of a set

.

Prove

is always open

Proof

Consider

.

a neighborhood

of

such that

This

is an open set, so

a neighborhood

of

so

So

consists of interior points of

, so

is open.

QED

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