How would I explain an 'open set' Looking for an intuitive explanation. 
 A: An open set is a set where every point has some "wiggle room" without leaving the set.  No matter which point in the set you pick there is a little bit of space around that point (in every direction) that is still in the set.  In other words, no point in the set is on a boundary (if you were on a boundary, you couldn't move at all in the direction that would take you across the boundary).
A: I think the idea of a boundary ($\partial A = \overline{A} \setminus A^\circ$) is very helpful.
Essentially, an open set doesn't contain any of its boundary; there is a demarcation object (I want to say point, but it's the boundary) separating our set, $A$, from everything that's not in $A$. Our set is open if this demarcation is completely outside $A$.
In other words, it's hard to tell (from within $A$) where $A$ ends. We never reach any point $x \in A$ for which we can say, "Aha, going beyond this point, I would no longer be in $A$!"
Looking at $A$ from the outside, we would encounter points for which we cannot continue "going toward $A$", while remaining outside $A$; we reach the edge of $X \setminus A$, while remaining in $X \setminus A$.
A: Intuitively, an open set is the set that does not include both endpoints.
For example, (0,1) is the open set; [0,1] is the close set, and (0,1] is neither open nor close.
