Prove that the intersection of any collection of closed sets in a topological space X

In summary: The set on the right hand side of the above equation is the complement of an open set. Hence the intersection F_i is closed.
  • #1
son
7
0
Prove that the intersection of any collection of closed sets in a
topological space X is closed.







Homework Statement


Homework Equations


The Attempt at a Solution


 
Physics news on Phys.org
  • #2


What have you tried?

Where are you stuck?

How is a closed set defined?
 
  • #3


think about complements
 
  • #4


I believe, the shortest and the easiest way is to start with the definition of a closed set and what an intersection is.

Good Luck.
 
  • #5


Indeed, the shortest and easiest way to prove anything is to start with the definitions!

However, there are many different ways to define "closed set". son, what definition are you using?
 
  • #6


the theorem i am using for a closed set is...
Let X be a topological space. the following statements about the collection of closed set in X hold:
(i) the empty set and X are closed
(ii) the intersection of any collection of closed sets is a closed set
(iii) the union of finitely many closed sets is a closed set
 
  • #7


I am new in the forum ( although I have read it for some time ) and I am not quite sure how much of a hint I am allowed to give you but:

You can prove that a set is closed using "balls". I would suggest you to work in this direction.
 
  • #8


son said:
the theorem i am using for a closed set is...
Let X be a topological space. the following statements about the collection of closed set in X hold:
(i) the empty set and X are closed
(ii) the intersection of any collection of closed sets is a closed set
(iii) the union of finitely many closed sets is a closed set

Yes, that's the theorem you are trying to prove. But what is the definition of a closed set??
 
  • #9


This follows directly from the definition. Please show us what part you're having trouble with. This is not a homework answer generator.
 
  • #10


A closed set is the complement of ___?___ .
 
  • #11


the definition of a closed set is... a subset A of a topological space X is closed if the set X - A is open.

but I am not sure how i would start the proof...
 
  • #12


the definition of a closed set is... a subset A of a topological space X is closed if the set X - A is open.

but I am not sure how i would start the proof...
 
  • #13


What is the definition of a topological space? Doesn't it say something about the union of a collection of open sets?
 
  • #14


son said:
the definition of a closed set is... a subset A of a topological space X is closed if the set X - A is open.

but I'm not sure how i would start the proof...
I would start it something like:

[tex]\text{Let }\left\{A_\alpha\right\}\text{ be an arbitrary collection of closed sets in a topological space }X, \text{ where }\alpha\in I,\ I \text{ an indexing set.}[/tex]

[tex]\text{The set }C=\bigcup_{\alpha\in I}\,A_\alpha\ \text{ is the union of an arbitrary collection of closed sets in }X.[/tex]

...

Now show that the compliment of set C is open in X.
 
  • #15


Let F={F_i} be a collection of closed sets. Then F_i=X-U_i for some collection {U_i} of open sets of X, because of the definition of closed. Then De-Morgans rules give
intersection F_i = intersection (X-U_i) = X - (union U_i)
But union U_i is an open set because the unions of open sets are open. Thus the set on the right hand side of the above equation is the complement of an open set. Hence the intersection F_i is closed.

does this look RIGHT??
 
  • #16


That's the general idea. It could be a bit more polished.
 
  • #17


culturedmath said:
I am new in the forum ( although I have read it for some time ) and I am not quite sure how much of a hint I am allowed to give you but:

You can prove that a set is closed using "balls". I would suggest you to work in this direction.
No, you cannot. That works only in a "metric space" because balls are only defined in a metric space. This problem clearly is about general topological spaces.
 
  • #18


Suppose that [itex]\{F_i|i\in I\}[/itex] is a collection of closed sets. You want to prove that

[tex]\bigcap_{i\in I}F_i[/tex]

is closed. By the definition you posted, this is the same thing as showing that

[tex]\Big(\bigcap_{i\in I}F_i\Big)^c[/tex]

is open. Can you think of a way to rewrite this last expression as something that's obviously open?
 

1. What is the definition of a closed set in a topological space?

A closed set in a topological space is a subset of the space that contains all of its limit points. In other words, a closed set includes all of the points that can be approached arbitrarily closely by points in the set itself.

2. What is the intersection of two closed sets in a topological space?

The intersection of two closed sets in a topological space is a new set that contains only the points that are common to both of the original sets. This new set is also a closed set, as it contains all of the limit points of the original sets.

3. Is the intersection of any collection of closed sets always a closed set?

Yes, the intersection of any collection of closed sets in a topological space is always a closed set. This is because the intersection contains only the points that are common to all of the original closed sets, making it a closed set itself.

4. How does the intersection of closed sets relate to the closure of a set in a topological space?

The intersection of any collection of closed sets in a topological space is a subset of the closure of that same set. In other words, the intersection is a "subset" of the closure, meaning it contains some, but not necessarily all, of the points in the closure.

5. Can the intersection of closed sets be empty?

Yes, it is possible for the intersection of closed sets in a topological space to be empty. This occurs when there are no points that are common to all of the original closed sets, meaning the intersection contains no points and is therefore an empty set.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
703
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
4K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
Back
Top