Question

If G is an open set and F is a closed set, then G \ F
is open and F \ G is closed. Show.

Answer 1

yes your answer is right

Answer 2

Let \(G\) be an open set, and let \(F\) be a closed set. Then\[G\setminus F=\left \{ x\in G:x\notin F \right \}.\]However, notice that for \(x\in G\setminus F\), \(x\in G\), which is an open set. Therefore, \(x\in G\setminus F\) must be open as well.

You can do the same for the other case.

Answer 3

I just stumbled over this question by chance. I think you need a topological argument to prove this. Let G be an open set and F be a closed set in some topology T on X. Then by definition of a closed set, X\F is open in T. Now \[G \setminus F = G \cap (X \setminus F)\]
By the definition of a topology the intersection of two open sets is an open set, too.
Similarly \[F \setminus G = F \cap (X \setminus G)\]
which is the intersection of two closed sets. It can be shown that the intersection of two closed sets is a closed set. I assume that proofing this is not what's being asked here. (See http://en.wikipedia.org/wiki/Closed_set: "Any intersection of closed sets is closed") \[\Box\]