2009-54.md

course	course_year	question_number	tags	title	year
Metric and Topological Spaces	IB	54	IB 2009 Metric and Topological Spaces	Paper 3, Section I, F	2009

Are the following statements true or false? Give brief justifications for your answers.

(i) If $X$ is a connected open subset of $\mathbb{R}^{n}$ for some $n$, then $X$ is path-connected.

(ii) A cartesian product of two connected spaces is connected.

(iii) If $X$ is a Hausdorff space and the only connected subsets of $X$ are singletons ${x}$, then $X$ is discrete.