int(A) + ext(A) not dense

[gonzo]

Can someone help me find an example of how the union of int(A) and ext(A) doesn't have to be dense in some space X? Thanks.

[morphism]

Define your terms. What are int(A) and ext(A)? The former I'm assuming is the interior of A, and the latter is ... the interior of the complement of A (= the complement of the closure of A)? And post your thoughts on the matter.

[gonzo]

int(A) = interior of A
ext(A) = exterior of A, or the interior of the complement of A

My thoughts are thus: int(A) and ext(A) are both open sets, so their union is an open set, and if we let B = union of int(A) and ext(A) then B = int(B). So the only way the closer is not equal to the entire space (making it dense) would be if the complement of B had some open points so that ext(B) was not empty.

However, since the complement of B is a subset of the complement of A, and ext(A) is all the open points of of the complement of A, the only way I can see that this would have a chance of being possible is if somehow you could construct a space where the int(A)=ext(A) for some set in that space (neither of which equaled the entire space). But I can't figure out how to construct a space where this is possible.

Those are my thoughts.

[HallsofIvy]

What in the world are "open points"? Do you mean "interior points" of a given set? In general topology, points do not have any properties- "points are points".

[gonzo]

Sorry, bad phrasing. I figured it should have been obvious what I meant from context. I believe my book calls them "interior points", which are points that are contained in some open set that is completely contained in the set in question.

[gonzo]

Nevermind, I got it. I forgot about the possibility that int(A) and ext(A) could both be the empty set, and thus the closure would also be the empty set (which I guess meets my criteria anyway of int(A) = ext(A)).