Interior of a set?

Question

I'm trying to think if their is any topology for which this is false:

If G is an open set, then G = interior(G)

Can anybody think of anything? I'm pretty sure it's straight forward.

Answer 1

From one of definitions, interior of a set is a largest open set contained in it. It means, that your statement ($\operatorname{Int}(G)=G$ for open $G$) is true.

Answer 2

By definition int(G) := $\bigcup_{U \subseteq G;\, U \text{open}} U = G$. In any topology.

Answer 3

It's true for any topology, and the proof depends on the form of your definition of a set's interior. For example, one definition of the interior of a set $A$ is as the union of all open sets $U$ such that $U\subseteq A.$ Another (equivalent) definition is that the interior of a set $A$ is the unique open set $U$ such that for every open set $V$ with $V\subseteq A,$ we have $V\subseteq U$. (The latter definition is sometimes expressed as "[T]he interior of a set $A$ is the largest open set contained in $A$.)

Regardless of the definition, it is provable. If you have any trouble with the proof, feel free to let me know. (I will need to know what definitions you're using, of course.)