Interior and Closure of set proof
I'm currently studying for a deferred final, and understand the general concept of this question, but was hoping I could get a clear answer from someone knowledgable in order to reinforce my knowledge.
The Question: Let S be a set in R^n. Explain what is meant by the interior S^o of S. If the closure of S^o is T = [S^o with a bar above it, I'm horrible with LaTeX], and S is a closed set, prove that T is a subset of S. Give an example of a closed set in S where the inclusion is strict
My attempted solution: Well, I don't totally have a solution. Here's what I know though. The interior of a set is all points in a set that are not on the boundary. That is, all points in S that for any delta > 0 there exists a neighborhood N such that N is an open set in S. I feel my definition is weak, and I don't totally understand what a closure is and thus cannot go much further with this
If someone could help me with this question I'd greatly appreciate it.