Due date: August 25
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 have the same convergent sequences. Then is compact if and only if is compact.
Let be a Cauchy sequence and . Then .
Give necessary and sufficient conditions for the discrete space to be compact.
- Let be a bounded sequence in the Euclidean space . Then it has a convergent subsequence (assume the result in ).
- A closed rectangle in is compact.
- A closed ball in is compact.