| course |
course_year |
question_number |
tags |
title |
year |
Metric and Topological Spaces |
IB |
56 |
IB |
2010 |
Metric and Topological Spaces |
|
Paper 1, Section II, H |
2010 |
Let $f: X \rightarrow Y$ and $g: Y \rightarrow X$ be continuous maps of topological spaces with $f \circ g=\mathrm{id}_{Y}$.
(1) Suppose that (i) $Y$ is path-connected, and (ii) for every $y \in Y$, its inverse image $f^{-1}(y)$ is path-connected. Prove that $X$ is path-connected.
(2) Prove the same statement when "path-connected" is everywhere replaced by "connected".