Due date: August 25
Problem 1
Let X be a compact space and $latex f:X\to Y$ a continuous bijection. Then $latex f^{-1}:Y\to X$ is continuous.
Give an example of a continuous bijection $latex f:X\to Y$, for a noncompact X, whose inverse is not continuous.
Problem 2
Let $latex (X,d_1), (X,d_2)$ have the same convergent sequences. Then $latex (X,d_1)$ is compact if and only if $latex (X,d_2)$ is compact.
Problem 3
Let $latex x_n$ be a Cauchy sequence and $latex x_{n_k}\to x$. Then $latex x_n \to x$.
Problem 4
Give necessary and sufficient conditions for the discrete space $latex (X,d)$ to be compact.
Problem 5
- Let $latex x_k$ be a bounded sequence in the Euclidean space $latex \R^n$. Then it has a convergent subsequence (assume the result in $latex n=1$).
- A closed rectangle in $latex \R^n$ is compact.
- A closed ball in $latex \R^n$ is compact.
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