1. ## 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)

2. ## 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.