Basic topology: closure and union

In summary, the author is having trouble proving that the closure of a set is equal to itself union its limit points. The contrapositive is easy to show, and if x is not in Cl(S)∪Cl(T), then it cannot be in Cl(S∪T).
  • #1
arty21
1
0

Homework Statement


[itex]Cl(S \cup T)= Cl(S) \cup Cl(T)[/itex]

Homework Equations


I'm using the fact that the closure of a set is equal to itself union its limit points.

The Attempt at a Solution


I am just having trouble with showing [itex]Cl(S \cup T) \subset Cl(S) \cup Cl(T)[/itex]. I can prove this one way, but I figured out another way that I prefer and want to know if there is any flaw in it:

Basically it boils down to showing the limit points of [itex]S \cup T[/itex] is a subset of the limit points of S union the limit points of T. So, if x is a limit point of [itex]S \cup T[/itex] then every open set U containing x intersects [itex]S \cup T[/itex] in a point other than x. Hence, x is a limit point of S or T.

I keep going back and forth. Sometimes I feel like this is fine and other times I feel like I'm making an error because if x is a limit point of S this means that every neighborhood of x intersects x in a point other than x. But in the proof I just gave, we know every neighborhood of x intersects [itex]S \cup T[/itex] in a point other than x, so we don't know if it necessarily always in S or T. I don't know why I'm having so much trouble with this! I think I am just over thinking it or something. So is this proof fine then?
 
Last edited:
Physics news on Phys.org
  • #2
Well, I think you've figured out exactly what the problem is. And it is a problem. It's good to have a feeling for when a proof is incomplete. You haven't proved it this way.
 
  • #3
I'm not certain this can be proved directly like this, but the contrapositive is pretty easy to show. If x is not in Cl(S)∪Cl(T), then show it can't be in Cl(S∪T).
 

What is closure in basic topology?

Closure in basic topology refers to the set of all points that are either in a given set or are limit points of that set. In other words, it is the smallest closed set that contains the given set.

What is the union of sets in basic topology?

In basic topology, the union of two or more sets is the set that contains all elements that are in at least one of the given sets. It is denoted by the symbol ∪.

Can the closure of a set be empty?

Yes, it is possible for the closure of a set to be empty. This happens when the set itself is empty or when all of its elements are isolated points, meaning they have no limit points.

How do you find the closure of a set in basic topology?

To find the closure of a set in basic topology, you can take the set itself and add all of its limit points to it. Alternatively, you can take the intersection of all closed sets that contain the given set.

What is the relationship between closure and union in basic topology?

The closure of a set is equal to the union of the set and its limit points. In other words, the closure of a set is the union of the set with its boundary.

Similar threads

  • Topology and Analysis
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
849
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
938
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
3K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Back
Top