Due September 21
Let X be a compact space and a continuous bijection. Then is continuous.
Give an example of a continuous bijection , for a noncompact X, whose inverse is not continuous.
Let X be a discrete metric space.
- Any function on X is uniformly continuos.
- Is X always compact?
- If not, give necessary and sufficient conditions for a discrete space to be compact.
Let X be a subspace of Y, i.e. a subset of the metric space Y with the same metric. We say X is closed in Y if X contains all its limits: if is a sequence that converges to , then .
If Y is complete, then X is complete if and only if X is closed in Y.
- Every bounded set in is totally bounded.
- Every closed and bounded set in is compact.