If A is closed in B, and B closed in C, what happens with A in C?

Question

If A is a closed subset of B, and B is a closed subset of C, does this imply that A is closed in C?

Answer 1

Yes! Since A is a closed subset of B it means that all closure points of A are in A for the topology in B, but there may be closure points of A that are not in A for the topology in C, but since B is closed in C (so every closure point of B is in B) it implies that every closure point of A is in A with respect to topology in C.

Example: Let A=[3,4]; B=[2,5]; C=[1,6]. Than A is closed in B, B is closed in C and A is closed in C. If we had P=[3,4); Q=[2,4) and R=[1,5] than P is closed in Q, but Q is not closed in R, so P is not closed in R.