Set theory/topology proof

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In summary, given a set K in G that is closed and a set F in K that is closed, the task is to prove that F is closed in G. By showing that F's complement, which is equal to H intersect K, is open, it is proven that F is closed in G. This is based on the fact that the topology on K is the topology of G restricted to K, meaning every open set in K is equal to a set open in G intersect K. Drawing a picture may be helpful in understanding the problem, but it is not necessary to derive the solution.
  • #1
i want to prove that given a set K in G that is closed and a set F in K that is closed, then K is closed in G. K, F, and G are all topological spaces.

so to reword the problem i instead want to show that given G-K is open in G and K-F is open in K show that G-F is open in G. so since G is a topological space and K is a subspace of G, i endow K with the subspace topology. then there exists some open set O in G such that any open subset of K such as K-F = O ∩ K. then F = K - (O∩K) then i drew myself a picture and found that F also equals (G-O)∩K which works because G-F = G-(G-O) ∪ (G-K) = O U (G-K) and since O is open in G and (G-K) is open in G therefore the union of those 2 sets is also open in G and G-F is then open in G which means F is closed in G.

i wish to know that in these types of proofs, is it common to have to draw pictures in order to continue? i was stuck at F = K - (O∩K) for a while until i drew a picture and saw that F = (G-O)∩K which i could take the complement of since it was a subset of G. my question is could F = (G-O)∩K be derived from F = K - (O∩K) or any of the other information i was given, without having to draw pictures to derive it? or is drawing pictures the usual way of going through these set theory types of proofs. thanks.
 
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  • #2
demonelite123 said:
i want to prove that given a set K in G that is closed and a set F in K that is closed, then K is closed in G. K, F, and G are all topological spaces.
so haven't you said K is closed in G already? and you want to show K is closed in G?

do you mean:
- F is closed in K
- K is closed in G
show F is closed in G
 
  • #3
oh whoops. i made a typo there. yes you are correct. if K in G is closed and F in K is closed i want to show that F in G is closed.
 
  • #4
The most direct way to show F is closed in G is to show that its complement is open. Let H be the complement of F in G. Of course, The complement of F in K is just H intersect K.

The topology on K is the topology of G restricted to K. That is, every open set in K is equal to a set open in G intersect K.
 
  • #5
thanks! your proof was a lot more concise than mines.
 

1. What is set theory and topology?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects. Topology is a branch of mathematics that studies the properties of space and the relationships between points in that space.

2. What is a proof in set theory/topology?

A proof in set theory/topology is a logical argument that uses axioms, definitions, and previously proven theorems to demonstrate the truth of a statement or proposition.

3. How do you construct a proof in set theory/topology?

To construct a proof in set theory/topology, one must first clearly state the statement to be proven, then use logical reasoning to break down the statement into smaller parts and show how each part can be deduced from the axioms, definitions, and previously proven theorems. The proof must be rigorous and follow a clear and logical structure.

4. What are some common techniques used in set theory/topology proofs?

Some common techniques used in set theory/topology proofs include direct proof, proof by contrapositive, proof by contradiction, and proof by induction. Other techniques include using properties of sets such as unions, intersections, and complements, as well as using concepts from other fields of mathematics such as algebra and calculus.

5. Can you give an example of a famous set theory/topology proof?

One famous proof in set theory/topology is the proof of the Cantor-Bernstein-Schröder Theorem, which states that if there exist injective functions from set A to set B and from set B to set A, then there exists a bijective function between the two sets. This proof uses the concept of a bijection and properties of sets such as unions and intersections.

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