Topology

Feb 2014
48
3
USA
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)
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
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 \(\displaystyle C\subseteq A\cup B\), p is in A only, or p is in B only, or p is in \(\displaystyle A\cap B\).

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

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