Closed sets in a topological space

In summary, the question posed is whether any closed subset of A is also a closed subset of B, given that both A and B are subsets of a topological space (X, tau). The answer is no, as A may not be closed as a subset of B. However, if A is compact in B, then the statement is true. Compactness does not imply closedness, as seen in the example of the Zariski topology on R.
  • #1
logarithmic
107
0
If [tex]A\subseteq B[/tex] are both subsets of a topological space [tex](X,\tau)[/tex], is it true that any closed subset of A is also a closed subset of B?
 
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  • #2
logarithmic said:
If [tex]A\subseteq B[/tex] are both subsets of a topological space [tex](X,\tau)[/tex], is it true that any closed subset of A is also a closed subset of B?

Note that A is a closed subset of A...
 
  • #3
Hurkyl said:
Note that A is a closed subset of A...
Hmm, I'm not sure. I never said A was closed. A is in B which is in X. And some other set in A, maybe call it U, is closed. I think the answer would be U is closed in X but I don't quite see why.
 
  • #4
Hurkyl's point is that any topological space is both open and closed as a subset of itself. If A is NOT closed as a subset of topological space B, since it IS closed as a subset of itself, the statement "any closed subset of A is also a closed subset of B" is false.
 
  • #5
HallsofIvy said:
Hurkyl's point is that any topological space is both open and closed as a subset of itself. If A is NOT closed as a subset of topological space B, since it IS closed as a subset of itself, the statement "any closed subset of A is also a closed subset of B" is false.
Ahh i see. Thanks. But is that the only problem here? If we insist that A is compact in B, then that fixes the problem and the statement is true, right?
 
  • #6
Um, just insisting that A is closed is all you need, nothing to do with compactness. In fact compactness won't help you at all - compact does not imply closed (e.g. the Zariski topology on R).
 
  • #7
?? I thought compact did imply closed!

Oh, I see. I started to give the proof and then realized I was saying "given points p and q construct neighborhoods about p and q that do not intersect". That's not possible in some topological spaces.
 
  • #8
If you don't know what the Zariski topology is (and Halls does but forget, temporarily) consider the topology on R given by:

U is open if and only if U contains the interval (0,1) - the set (0,1) is in this and is certainly compact, but not closed.
 

1. What is a closed set in a topological space?

A closed set in a topological space is a set that contains all of its limit points. This means that any sequence of points within the set that converges to a point outside of the set, is not considered a limit point. In simpler terms, a closed set is a set that includes its boundary points.

2. How is a closed set different from an open set?

An open set in a topological space is a set that does not contain its boundary points. This means that any point within the set has a neighborhood that is entirely contained within the set. Closed sets, on the other hand, include their boundary points. This difference is important when studying continuity and compactness in topological spaces.

3. Can a set be both open and closed in a topological space?

In general, a set cannot be both open and closed in a topological space. However, in certain topological spaces, such as the discrete topology, every subset is both open and closed. In most cases, though, a set is either open or closed, but not both.

4. How are closed sets related to the complement of open sets?

The complement of an open set in a topological space is a closed set. This means that any point in the complement is a boundary point of the open set. Similarly, the complement of a closed set is an open set. This relationship is known as the topological definition of closed sets.

5. What is the importance of closed sets in topology?

Closed sets play a crucial role in topology as they are used to define many important concepts, such as continuity, compactness, and connectedness. They also have applications in various fields, including mathematics, physics, and computer science. Understanding closed sets is essential for studying topological spaces and their properties.

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