Section 5 Metric Spaces
Definition 5.1.
Let \(d:X^2\to[0,\infty)\) be a function satisfying the following for all \(x,y,z\in X\text{.}\)
\(d(x,y)=0\) if and only if \(x=y\text{.}\)
\(\displaystyle d(x,y)=d(y,x)\)
\(\displaystyle d(x,z)\leq d(x,y)+d(y,z)\)
Then \(d\) is said to be a metric on the set \(X\text{,}\) and
\begin{equation*}
B_r(x)=\setBuilder{y\in X}{d(x,y)\lt r}
\end{equation*}
is said to be a metric ball around \(x\).
Checkpoint 5.2. Examples of metrics.
Verify that each of the following is a metric.
\(d(x,y)=1\) for all distinct \(x,y\in X\text{,}\) and \(d(x,x)=0\)
\(d(x,y)=|y-x|\) for all \(x,y\in\mb R\)
\(d(\tuple{x_0,x_1},\tuple{y_0,y_1})=\sqrt{(y_1-y_0)^2+(x_1-x_0)^2}\) for all \(\tuple{x_0,x_1},\tuple{y_0,y_1}\in\mb R^2\text{.}\)
\(d(\tuple{x_0,x_1},\tuple{y_0,y_1})=|y_1-y_0|+|x_1-x_0|\) for all \(\tuple{x_0,x_1},\tuple{y_0,y_1}\in\mb R^2\text{.}\)
\(d(\tuple{x_0,x_1},\tuple{y_0,y_1})=\max\setList{|y_1-y_0|,|x_1-x_0|}\) for all \(\tuple{x_0,x_1},\tuple{y_0,y_1}\in\mb R^2\text{.}\)
Theorem 5.3.
Let \(d\) be a metric on a set \(X\text{.}\) Then
\begin{equation*}
\mc B=\setBuilder{B_r(x)}{x\in X,r>0}
\end{equation*}
is a basis for a topology on \(X\text{.}\)
Definition 5.4.
The topology generated by the basis given in
Theorem 5.3 is called the
topology generated by the metric.
A given topology is said to be metrizable if there exists some metric that generates it. Two metrics are said to be topologically equivalent if they generate the same topology.
Proposition 5.5.
Every discrete space is metrizable.
Theorem 5.6.
Every metrizable space is \(T_4\text{.}\)
Theorem 5.7.
Let \(B_{r}(x),B_{r}'(x)\) be the metric balls around \(x\) given by two metrics \(d,d'\) respectively. Then \(d,d'\) are topologically equivalent if and only if for all \(x\in X\) and \(\epsilon\gt 0\text{,}\) there exists \(\delta\gt 0\) such that \(B_{\delta}(x)\subseteq B_{\epsilon}'(x)\) and \(B_{\delta}(x)'\subseteq B_{\epsilon}(x)\text{.}\)
Definition 5.8.
For \(\vec x=\tuple{x_0,\dots,x_{n-1}}\in\mb R^n\text{,}\) let \(\vec x(i)=x_i\text{.}\)
Theorem 5.9.
The following metrics on \(\mb R^n\) are topologically equivalent.
\(\displaystyle d(\vec x,\vec y)=\sqrt{\sum_{0\leq i\lt n}(\vec y(i)-\vec x(i))^2}\)
\(\displaystyle d(\vec x,\vec y)=\sum_{0\leq i\lt n}|\vec y(i)-\vec x(i)|\)
\(\displaystyle d(\vec x,\vec y)=\max\setBuilder{|\vec y(i)-\vec x(i)|}{0\leq i\lt n}\)
Definition 5.10.
The topology generated by the metrics given in
Theorem 5.9 is called the
Euclidean topology on
\(\mb R^n\text{.}\)
Proposition 5.11.
Definition 5.12.
A local basis at a point \(x\) is a collection of open sets \(\mc B_x\) such that for every neighborhood \(U\) of \(x\text{,}\) there exists \(B\in\mc B_x\) such that \(x\in B_x\subseteq U\text{.}\)
Definition 5.13.
A space is said to have local countable weight (or be first-countable) if there exists a countable local basis at every point of the space.
Proposition 5.14.
Every space of countable weight has local countable weight.
Proposition 5.15.
Every metrizable space has local countable weight
Theorem 5.16.
Let \(X\) be metrizable. Then \(X\) has countable weight if and only if it has countable density.
Proposition 5.17.
Every space \(\mb R^n\) with the Euclidean topology has countable weight and density.
FALL 2022 note: we will not cover the below topics.
Theorem 5.18.
For \(\vec x,\vec y\in\mb R^2\text{,}\) let \(d(\vec x,\vec y)=1\) if \(\vec x(1)\not=\vec y(1)\text{,}\) and \(d(\vec x,\vec y)=|\vec y(0)-\vec x(0)|\) otherwise. Then \(d\) is a metric generating a non-discrete topology on \(\mb R^2\) with uncountable weight and uncountable density.
Theorem 5.19.
The subspace
\(\setBuilder{\vec x}{\vec{x}(1)\in\setList{0,1}}\) of the space defined in
Theorem 5.18 is homeomorphic to the subspace
\((0,1)\cup(2,3)\) of the Euclidean line.
Definition 5.20.
A point \(x\) is called a sequential limit point of a set \(A\) iff there exists a countable subset \(B\subseteq A\setminus\setList{x}\) such that every neighborhood of \(x\) contains all but finitely many points of \(B\text{.}\)
Proposition 5.21.
Every sequential limit point of a set is a limit point of that set.
Theorem 5.22.
Let \(X\) have local countable weight. Then \(x\) is a limit point of a set if and only if \(x\) is a sequental limit point of that set.
Definition 5.23.
A Cauchy sequence is a countably infinte set \(A\) such that for all \(\epsilon\gt0\text{,}\) the set \(\setBuilder{x\in A}{\exists y\in A(d(x,y)\geq\epsilon)}\) is finite.
Definition 5.24.
A complete metric is a metric such that every Cauchy sequence has a sequential limit point.
A topology that can be generated by a complete metric is said to be completely metrizable.
Proposition 5.25.
Every Euclidean space is completely metrizable.
Proposition 5.26.
Let \(d:X\to[0,\infty)\) be a metric and \(Y\subseteq X\text{.}\) Then \(d\) restricted to \(Y\) generates the subspace topology on \(Y\text{.}\) (Therefore, every subspace of a metrizable space is metrizable.)
Theorem 5.27.
The subspace \((0,1)\) of the Euclidean line is completely metrizable, but not by the topology inherited from \(\mb R\text{.}\)
Theorem 5.28.
The subspace \(\mb Q\) of the Euclidean line is metrizable, but not completely metrizable.
Theorem 5.29.
The subspace \(\mb R\setminus\mb Q\) of the Euclidean line is completely metrizable, but not by the topology inherited from \(\mb R\text{.}\)
Theorem 5.30.
Metrizable and completely metrizable are topological properties. That is, if \(X\) and \(Y\) are homeomorphic, then \(X\) is (completely) metrizable if and only if \(Y\) is too.
