| course |
course_year |
question_number |
tags |
title |
year |
Metric and Topological Spaces |
IB |
56 |
IB |
2015 |
Metric and Topological Spaces |
|
Paper 1, Section II, E |
2015 |
Give the definition of a metric on a set $X$ and explain how this defines a topology on $X$.
Suppose $(X, d)$ is a metric space and $U$ is an open set in $X$. Let $x, y \in X$ and $\epsilon>0$ such that the open ball $B_{\epsilon}(y) \subseteq U$ and $x \in B_{\epsilon / 2}(y)$. Prove that $y \in B_{\epsilon / 2}(x) \subseteq U$.
Explain what it means (i) for a set $S$ to be dense in $X$, (ii) to say $\mathcal{B}$ is a base for a topology $\mathcal{T}$.
Prove that any metric space which contains a countable dense set has a countable basis.