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Topology Proof (Closed/Open Sets)

  • Thread starter tylerc1991
  • Start date
  • #1
166
0

Homework Statement


Let (X,T) be a topological space,
let C be a closed subset of X,
let U be an open subset of X.

Prove that C - U is closed and U - C is open.

The Attempt at a Solution


I was trying to do this by 4 cases:

Case 1: Let U be a proper subset of C.
Then U - C = empty and hence is open.

Case 2: Let C be a proper subset of U.
Then C - U is empty and hence is closed.

Case 3: Let U be a proper subset of C.
Then C - U = ?

Case 4: Let C be a proper subset of U.
Then U - C = ?

Cases 3 and 4 are where I am stuck. Can someone give me some intuition to get me started on them? Thank you!
 

Answers and Replies

  • #2
lanedance
Homework Helper
3,304
2
I don't thinnk you need cases, I would start by writing the difference term using complements
[tex] C - U = C \cap U^c [/tex]
 

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