2017-56.md

course	course_year	question_number	tags	title	year
Metric and Topological Spaces	IB	56	IB 2017 Metric and Topological Spaces	Paper 1, Section II, E	2017

Consider $\mathbb{R}$ and $\mathbb{R}^{2}$ with their usual Euclidean topologies.

(a) Show that a non-empty subset of $\mathbb{R}$ is connected if and only if it is an interval. Find a compact subset $K \subset \mathbb{R}$ for which $\mathbb{R} \backslash K$ has infinitely many connected components.

(b) Let $T$ be a countable subset of $\mathbb{R}^{2}$. Show that $\mathbb{R}^{2} \backslash T$ is path-connected. Deduce that $\mathbb{R}^{2}$ is not homeomorphic to $\mathbb{R}$.