# Thread: Prove that C is closed in X

1. ## Prove that C is closed in X

Could someone help me with the following problem? Any help would be greatly appreciated.

Let X be a metric space, and let Y c X be closed (in X) and C c Y be closed in Y. Prove that C is closed in X. Show by example that C may or may not be closed if Y is not closed in X.

Thanks

2. Let $\displaystyle p \in Y$ be a Limit Point of C that is not in Y. (If there is no such point, we are done since C is then closed in X because C is closed in Y.) Since p is a Limit point there are infinitly many points of C in any neighbourhood of p. Since C is a subset of Y, there are also infinitely many points of Y in this neighbourhood and thus, p is also a limit point of Y. Since Y is closed in X, p must be in Y. This is a contradiction to our assumption. Thus any limit point of C must be in C if C is closed in Y and Y is closed in X.

If Y is not closed in X, e.g.:

$\displaystyle C = [0,1), Y = [-1,1), X = R$ (the real numbers) then C is closed in Y but Y is not closed in X. Also C is not closed in X.

3. thanks i understand it now. good explanation.