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Thread: Topology

  1. #1
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    Topology

    Let A, B and C be subsets of a topological space X with C⊂A ∪ B. If A, B and A ∪ B are given the relative topologies, prove that C is open with respect to A ∪ B if and only if C∩A is open with respect to A and C∩B is open with respect to B.

    I need help with the part "if C∩A is open with respect to A and C∩B is open with respect to B then C is open with respect to A ∪ B"
    I already did the other direction. This is what I gathered, but have not had much luck connecting everything.
    We know that since C∩A is open with respect to A then C∩A=G∩A for some open set G on the topological space. Similarly, since C∩B is open with respect to B, then C∩B=H∩B for some open set H on the topological space. I tried using G∪H as my open set, and I cant seem to prove that (G∪H)∩(A ∪ B)=C, but to be honest I have tried to come up with something else, and have ran out of luck. How do I prove the existence of an open set N such that N∩(A ∪ B)=C?

    Thanks for any help in advance

    (This exercise is taken from Schaum's Outlines of General topology chapter 5 problem 90)
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  2. #2
    MHF Contributor

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    Re: Topology

    It's been a long time since I did anything in topology but since no one else has responded:

    "if C∩A is open with respect to A and C∩B is open with respect to B then C is open with respect to A ∪ B"
    Let p be a point in C. Since C\subseteq A\cup B, p is in A only, or p is in B only, or p is in A\cap B.

    1) p is in A only. Since C is open with respect to A, there exist an open set, U, such that U\cap A contains p. But then p is in U \cap (A\cup B).

    2) p is in B only. Change "A" to "B" in (1).

    3) p is in both A and B. There exist an open set, U, such that U\cap A contains p and there exist an open set, V, such that V\cap B contains p. Then W= U\cap V is an open set such that W\cap(A\cup B) contains p.
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