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Looking for an intuitive explanation.

Nishant S
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    "open set" is mathematical terminology, so you lost the game before starting it. –  Mar 11 '15 at 18:54
  • @Woodface - Agreed, edited. – Nishant S Mar 11 '15 at 18:55
  • Speaking very informally, an open set is one that is completely open for entry in the sense that it has no boundary points that might block your way when trying to enter the set. – Dave L. Renfro Mar 11 '15 at 19:03
  • @DaveL.Renfro - Sorry, that made little sense to me. – Nishant S Mar 11 '15 at 19:07
  • See the figure here, and notice that the boundary is dashed. Thus, no boundary points are included. I was thinking of boundary points as being points that you'd bump into when trying to get into the set from the outside, at least if you don't try to move around the boundary point. The fact that the word "boundary" also has this common use meaning (Merriam-Webster's Collegiate Dictionary, 11th edition: boundary -- something that indicates or fixes a limit or extent) was what I was playing around with. – Dave L. Renfro Mar 11 '15 at 20:08

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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).

TravisJ
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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 $X \setminus A$.

pjs36
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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.

MathnerdXjh
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