Is Int(A) + ext(A) dense in some cases?

  • Thread starter gonzo
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In summary, int(A) and ext(A) are both open sets, and their union is also an open set. However, in order for the union not to be dense in some space X, the complement of the union would have to have some open points. To achieve this, it is possible that int(A) and ext(A) could both be the empty set, resulting in the closure also being the empty set.
  • #1
gonzo
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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.
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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".
 
  • #5
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.
 
  • #6
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)).
 

What is the meaning of "Int(A) + ext(A) not dense"?

"Int(A) + ext(A) not dense" refers to a set A, where the interior points (Int(A)) and the exterior points (ext(A)) are not densely distributed. This means that there are gaps or spaces between the points, and the points are not closely packed together.

What is the significance of "Int(A) + ext(A) not dense" in mathematics?

The concept of "Int(A) + ext(A) not dense" is important in topology, as it helps to define the boundary of a set. The boundary of a set is the points that are neither in the interior nor the exterior of the set, and it is commonly denoted as ∂A. In other words, ∂A = Int(A) + ext(A) not dense.

Can a set have both dense and not dense interior and exterior points?

Yes, it is possible for a set to have both dense and not dense interior and exterior points. For example, a set can have a dense interior but not dense exterior, or vice versa. This depends on the distribution of points within the set and can vary for different sets.

What is the difference between "Int(A) + ext(A) not dense" and "Int(A) + ext(A) = ∂A"?

The main difference between these two expressions is that "Int(A) + ext(A) not dense" refers to the distribution of points within a set, while "Int(A) + ext(A) = ∂A" refers to the boundary of a set. In other words, "Int(A) + ext(A) not dense" describes the internal structure of a set, while "Int(A) + ext(A) = ∂A" defines the separation between a set and its surroundings.

Can "Int(A) + ext(A) not dense" be applied to higher dimensions?

Yes, the concept of "Int(A) + ext(A) not dense" can be extended to higher dimensions. In two or more dimensions, the interior points refer to the points inside the set, while the exterior points refer to the points outside the set. The same principle applies, where the points are not densely distributed.

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