Thanks, I think I've got it-
To show that U is open in X => U \cap S is open in S.
For any p \in U \cap S, U contains some open ball in X, say B_{U}, with center p. Say B_{U} has radius r. Let B_{S} be the open ball in S with the same center p and radius r. Then B_{S} is the set of all...
After some quick googling on subspace topologies:
The book I'm following (Introduction to Analysis, Maxwell Rosenlicht) doesn't refer to topological spaces in general, just metric spaces. It does define the subspace of a metric space as that metric space with a restricted underlying set and...
Hi, I'm having trouble understanding this proof.
Theorem. Let \{ S_{i} \} _{i \in I} be a collection of connected subsets of a metric space E. Suppose there exists i_{0} \in I such that for each i \in I, S_{i} \cap S_{i_{0}} \neq \emptyset.
Then \cup_{i \in I} S_{i} is connected.
Proof...