Section 20: The Metric Topology

General properties

• The topology induced by $d$ is the coarsest topology on $X$ such that $d:X\times X\rightarrow\mathbb{R}$ is continuous.
• The standard bounded metric corresponding to $d$ is $\overline{d}=\min\{d,1\}$ . $d$ and $\overline{d}$ induce the same topology.
  • Another example of a bounded metric inducing the same topology as $d$ is $d'=d/(1+d)$ .

Standard metrics on $\mathbb{R}^{J}$

• $d$ is the euclidean metric on $\mathbb{R}^{n}$ if $d(\mathbf{x},\mathbf{y})=||\mathbf{x}-\mathbf{y}||$ where $||\mathbf{z}||=\sqrt{\sum_{i=1}^{n}z_{i}^{2}}$ .
• $\rho$ is the square metric on $\mathbb{R}^{n}$ if $\rho(\mathbf{x},\mathbf{y})=\max_{i=1,...,n}\{|x_{i}-y_{i}|\}$ .
• $\overline{\rho}$ is the uniform metric on $\mathbb{R}^{J}$ if $\bar{\rho}(\mathbf{x},\mathbf{y})=\sup_{\alpha\in J}\{\overline{d}(x_{\alpha},y_{\alpha})\}$ .
  • The uniform metric induces the uniform topology.
  • For $0<\epsilon<1$ , $B_{\bar{\rho}}(\mathbf{x},\epsilon)=\cup_{\delta<\epsilon}U(\mathbf{x},\delta)$ where $U(\mathbf{x},\delta)=\prod_{\alpha\in J}(x_{\alpha}-\delta,x_{\alpha}+\delta)$ . $U(\mathbf{x},\delta)$ is open in the box topology but not in the uniform topology.
• $D(\mathbf{x},\mathbf{y})=\sup_{i\in\mathbb{Z}_{+}}\{\overline{d}(x_{i},y_{i})/i\}$ is a metric that induces the product topology on $\mathbb{R}^{\omega}$ .

Properties

• The euclidean and square metrics induce the standard topology on $\mathbb{R}^{n}$ .
  • And so does every metric of the form $(\sum_{i=1}^{n}|x_{i}-y_{i}|^{p})^{1/p}$ for $p\ge1$ .
• The box topology on $\mathbb{R}^{J}$ is finer than the uniform topology which is finer than the product topology. If $J$ is infinite, all three are different.
• $h((x_{1},x_{2},\ldots))=(a_{1}x_{1}+b_{1},a_{2}x_{2}+b_{2},\ldots)$ , $a_{i}>0$ for all $i\in\mathbb{Z}_{+}$ , which is a homeomorphism relative to the box and product topologies, is continuous relative to the uniform topology iff $a_{i}$ ’s are bounded from above, and is a homeomorphism iff $a_{i}$ ’s are bounded from below and above by some positive numbers.

Metrization of $\mathbb{R}^{J}$

• The metric $d$ is one that induces the product (box and uniform) topology on $\mathbb{R}^{n}$ .
• The metric $D$ is one that induces the product topology on $\mathbb{R}^{\omega}$ .
• As we shall see in §21, if $x\in\overline{A}\subset X$ and $X$ is metrizable, then there is a sequence of elements of $A$ converging to $x$ .
  • $\mathbb{R}^{\omega}$ in the box topology is not metrizable. If $A=\{\mathbf{x}|x_{i}>0\}$ then $\mathbf{0}\in\overline{A}$ in the box topology, but there is clearly no sequence of elements of $A$ converging to $\mathbf{0}$ in the box topology. A similar argument works for all infinite $J$ .
  • $\mathbb{R}^{J}$ in the product topology is not metrizable if $J$ is uncountable. If $A=\{\mathbf{x}|x_{\alpha}=1\mbox{ for all but finitely many values of }\alpha\}$ , then $\mathbf{0}\in\overline{A}$ in the product topology, but, if $J$ is uncountable, then for any sequence $\{\mathbf{x}_{n}\}$ of points of $A$ there is some $\beta$ such that $x_{n\beta}=1$ for all $n$ , hence, $\mathbf{x}_{n}$ does not converge to $\mathbf{0}$ in the product topology.

Subspaces of $\mathbb{R}^{\omega}$

• $\overline{\mathbb{R}^{\infty}}\subset\mathbb{R}^{\omega}$ equals $\mathbb{R}^{\infty}$ in the box topology, the set of all sequences of real numbers converging to $0$ in the uniform topology, and $\mathbb{R}^{\omega}$ in the product topology.
• The $l^{2}$ -topology is strictly finer than the uniform topology on $\mathbb{R}_{l^{2}}^{\omega}$ , but strictly coarser than the box topology (both inherited from $\mathbb{R}^{\omega}$ ).
• When all four (box, $l^{2}$ , uniform, and product) topologies are inherited by $\mathbb{R}^{\infty}$ , they are all distinct.
• When all four (box, $l^{2}$ , uniform, and product) topologies are inherited by the Hilbert cube, $H=\prod_{n\in\mathbb{Z}_{+}}[0,1/n]$ , the box topology is strictly finer than the other three topologies which become equal.