| course |
course_year |
question_number |
tags |
title |
year |
Metric and Topological Spaces |
IB |
56 |
IB |
2014 |
Metric and Topological Spaces |
|
Paper 1, Section II, E |
2014 |
Define what it means for a topological space to be compact. Define what it means for a topological space to be Hausdorff.
Prove that a compact subspace of a Hausdorff space is closed. Hence prove that if $C_{1}$ and $C_{2}$ are compact subspaces of a Hausdorff space $X$ then $C_{1} \cap C_{2}$ is compact.
A subset $U$ of $\mathbb{R}$ is open in the cocountable topology if $U$ is empty or its complement in $\mathbb{R}$ is countable. Is $\mathbb{R}$ Hausdorff in the cocountable topology? Which subsets of $\mathbb{R}$ are compact in the cocountable topology?