| course |
course_year |
question_number |
tags |
title |
year |
Analysis and Topology |
IB |
2 |
IB |
2021 |
Analysis and Topology |
|
Paper 1, Section II, F |
2021 |
Let $f: X \rightarrow Y$ be a map between metric spaces. Prove that the following two statements are equivalent:
(i) $f^{-1}(A) \subset X$ is open whenever $A \subset Y$ is open.
(ii) $f\left(x_{n}\right) \rightarrow f(a)$ for any sequence $x_{n} \rightarrow a$.
For $f: X \rightarrow Y$ as above, determine which of the following statements are always true and which may be false, giving a proof or a counterexample as appropriate.
(a) If $X$ is compact and $f$ is continuous, then $f$ is uniformly continuous.
(b) If $X$ is compact and $f$ is continuous, then $Y$ is compact.
(c) If $X$ is connected, $f$ is continuous and $f(X)$ is dense in $Y$, then $Y$ is connected.
(d) If the set ${(x, y) \in X \times Y: y=f(x)}$ is closed in $X \times Y$ and $Y$ is compact, then $f$ is continuous.