2021-1.md

course	course_year	question_number	tags	title	year
Analysis and Topology	IB	1	IB 2021 Analysis and Topology	Paper 4, Section I, $2 F$	2021

Let $X$ be a topological space with an equivalence relation, $\tilde{X}$ the set of equivalence classes, $\pi: X \rightarrow \tilde{X}$, the quotient map taking a point in $X$ to its equivalence class.

(a) Define the quotient topology on $\tilde{X}$ and check it is a topology.

(b) Prove that if $Y$ is a topological space, a map $f: \tilde{X} \rightarrow Y$ is continuous if and only if $f \circ \pi$ is continuous.

(c) If $X$ is Hausdorff, is it true that $\tilde{X}$ is also Hausdorff? Justify your answer.