Can open sets and closures intersect in a topological space?

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In summary, the conversation discusses whether the intersection of an open set A and the closure of another set B, where A and B are open sets in a topological Hausdorff space X, can be proven to be empty. It is shown that this is indeed the case, regardless of whether or not X is Hausdorff. The conversation also briefly mentions a potential counter-example, but it is ultimately deemed invalid.
  • #1
Shaji D R
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Suppose A and B are open sets in a topological Hausdorff space X.Suppose A intersection B is an empty set. Can we prove that A intersection with closure of B is also empty? Is "Hausdorff" condition necessary for that?

Please help.
 
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  • #2
Shaji D R said:
Suppose A and B are open sets in a topological Hausdorff space X.Suppose A intersection B is an empty set. Can we prove that A intersection with closure of B is also empty? Is "Hausdorff" condition necessary for that?

Please help.

Given that [itex]A[/itex] and [itex]B[/itex] are disjoint, the only way [itex]A[/itex] can intersect with the closure of [itex]B[/itex] is if there exists [itex]a \in A[/itex] such that [itex]a[/itex] is a limit point of [itex]B[/itex].

But that's impossible: [itex]A[/itex] is an open neighbourhood of [itex]a[/itex] which contains no points in [itex]B[/itex]. Hence [itex]a[/itex] is not a limit point of [itex]B[/itex].

This holds whether or not [itex]X[/itex] is Hausdorff.
 
  • #3
Rephrasing pasmith's argument without reference to individual points...

Let [itex]X[/itex] be any topological space, and suppose [itex]A,B\subseteq X[/itex] are disjoint and [itex]A[/itex] is open. Then:
-[itex]X\setminus A[/itex] is closed because [itex]A[/itex] is open.
-[itex]X\setminus A \supseteq B[/itex] because [itex]A,B[/itex] are disjoint.
-As the closure of [itex]B[/itex], the set [itex]\bar B[/itex] is the smallest closed set that contains [itex]B[/itex].
-In particular, [itex]\bar B \subseteq X \setminus A[/itex].
Rephrasing the last point, [itex]A,\bar B[/itex] are disjoint sets.
 
  • #4
Thank you very much
 
  • #5
A \cap B empty

If A and B are disjoint, the B is a subset of the complement of A.

If A is open, its complement is closed.

Hence, in this case, the closure of B is contained in the complement of A.

Hence, A and the closure of B are disjoint.

There is no need for the ambient space to be Hausdorff. There is no need for B to be ooen.
 
  • #6
ibdsm said:
If A and B are disjoint, the B is a subset of the complement of A.

If A is open, its complement is closed.

Hence, in this case, the closure of B is contained in the complement of A.

Hence, A and the closure of B are disjoint.

There is no need for the ambient space to be Hausdorff. There is no need for B to be ooen.

The space has to be Hausdorff and B has to be open. A = (0,1) and B = [1,0] is a counter example to what you say
 
  • #7
1. [1,0] is the empty set, as there are no real numbers which are both at least 1 and at most 0. Hence your "counter-example" fails.

2. By definition,
(i) a subset of a topological space is closed if and only if it is the complement of an open set
(ii) the closure of a subset, B, of a topological space is the smallest closed subset of the space which contains B.

There is no question of being a Hausdorff space or even a T1 or T0 space for this.
 
  • #8
Wow, I can't believe I wrote that. I was dead tired and my counter-example I was trying to type out was A = (0,1), B = [1,2]. But even that fails.

My apologies, I'll try not to post when I'm half awake anymore lol.
 
  • #9
I'm glad I'm not the only one to blunder at times!
 

Related to Can open sets and closures intersect in a topological space?

1. What is the definition of an open set?

An open set is a set in a topological space that does not contain its boundary points. In other words, for any point in the set, there exists a small enough open ball around that point that is also contained in the set.

2. How do you determine if a set is open or closed?

A set is open if it does not contain its boundary points, and closed if it contains all of its boundary points. This can also be determined by looking at the complement of the set - if the complement is open, then the original set is closed.

3. What is the closure of a set?

The closure of a set is the smallest closed set that contains all of the points in the original set. It is denoted by the symbol "cl" and can also be thought of as the union of the original set and its boundary points.

4. How are open sets and closed sets related?

Open sets and closed sets are complementary concepts - a set can either be open, closed, or neither. A set is open if and only if its complement is closed, and vice versa.

5. Can a set be both open and closed?

Yes, in certain topological spaces, a set can be both open and closed. For example, in the discrete topology, all sets are both open and closed. In other topological spaces, a set can be both open and closed if and only if it is the entire space or the empty set.

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