The Timpson Lyceum · General Topology (Point Set Topology)

Properties of Metric Spaces & Isometries – Lecture 17 (Topology)

Overview

This lecture digs into what a metric buys you beyond a bare topology and proves several structural facts about metric spaces. We show that every metric space is Hausdorff, recast continuity between metric spaces in the familiar $\epsilon$ – $\delta$ language, and give a metric-based test for when one topology is finer than another. We then define the distance between sets (not just points), the notion of a bounded metric, and the surprising fact that every metric can be replaced by a bounded one that generates the same topology. The lecture closes with isometries — distance-preserving bijections — and a proof that the Euclidean and taxicab metrics on $\mathbb{R}^2$ , though they give the same topology, are not isometric.

Key Concepts

Metric (recap). A metric on a set $X$ is a function $d : X \times X \to \mathbb{R}$ satisfying, for all $x,y,z \in X$ :

Non-negativity / identity: $d(x,y) \ge 0$ , with $d(x,y) = 0$ iff $x = y$ .
Symmetry: $d(x,y) = d(y,x)$ .
Triangle inequality: $d(x,z) \le d(x,y) + d(y,z)$ (a detour can only lengthen a path).

The pair $(X,d)$ is a metric space. A metric is a stronger notion than a topology: it measures distance, not just proximity.

Metric topology. Given $(X,d)$ , the open balls $B_d(x,\epsilon) = \{\, y \in X : d(x,y) < \epsilon \,\}$ (for $x \in X$ , $\epsilon > 0$ ) form a basis $\beta$ . The topology they generate is the metric topology. On $\mathbb{R}^2$ the Euclidean metric gives round balls; the taxicab metric gives diamonds.

Open-set characterization (Theorem 1). $U$ is open in the metric topology iff for every $y \in U$ there exists $\delta > 0$ with $B_d(y,\delta) \subseteq U$ . (In general $U$ is open iff it is a union of open balls; this is the sharper "ball centered at the point" version used repeatedly below.)

Every metric space is Hausdorff. More precisely, the topology generated by any metric is Hausdorff: any two distinct points can be separated by disjoint open neighborhoods. A key consequence — a topology that is not Hausdorff cannot come from any metric, so not every topology is metrizable.

Continuity ( $\epsilon$ – $\delta$ for metric spaces). For metric spaces $(X,d_X)$ , $(Y,d_Y)$ and $f : X \to Y$ :

f \text{ is continuous} \iff \forall x\,\forall \epsilon>0\ \exists \delta>0 :\ f\bigl(B_{d_X}(x,\delta)\bigr) \subseteq B_{d_Y}(f(x),\epsilon).

This generalizes the calculus $\epsilon$ – $\delta$ definition (which is just the case of the Euclidean metric on $\mathbb{R}$ ) to any two metric spaces, possibly with different metrics. Topologically, "continuous" still means preimages of open sets are open.

Comparing topologies (finer / coarser). If $d$ and $d'$ induce topologies $\tau$ and $\tau'$ on the same set $X$ , then $\tau'$ is finer than $\tau$ ( $\tau \subseteq \tau'$ ) iff

\forall x\,\forall \epsilon>0\ \exists \delta>0 :\ B_{d'}(x,\delta) \subseteq B_d(x,\epsilon).

Finer = more (and smaller) open sets — "sand vs. rock." Note the flip: the subset goes one way for the topologies ( $\tau \subseteq \tau'$ ) and the opposite way for the balls (the primed ball sits inside the unprimed one). Corollary: the Euclidean, taxicab, and max metrics on $\mathbb{R}^2$ all generate the same topology, since each one's balls fit inside the others'.

Distance between sets. For $A, B \subseteq X$ ,

d(A,B) = \inf \{\, d(a,b) : a \in A,\ b \in B \,\}

— the greatest lower bound of all point-to-point distances. The infimum (not a minimum) is essential: for the open intervals $A=(-2,0)$ and $B=(1,3)$ there is no closest pair of points, yet $d(A,B)=1$ . This is not a metric on subsets: $d(A,B)$ can be $0$ for disjoint (e.g. tangent) sets, and it fails the triangle inequality.

Bounded metric. $A \subseteq X$ is bounded under $d$ if there exists $\mu > 0$ with $d(p,q) < \mu$ for all $p,q \in A$ . If all of $X$ is bounded, $d$ is a bounded metric. Boundedness depends on the metric, not just the set.

The standard bounded metric. Define $d'(x,y) = \min\{\, d(x,y),\ 1 \,\}$ . Then $d'$ is a metric, it is bounded (by $1$ ), and it induces the same topology as $d$ . The cutoff $1$ is arbitrary; any positive constant works. The two metrics agree on small scales (where $d < 1$ ), and topology only cares about small scales — so they declare the same points "close" and generate the same open sets.

Isometry. A bijection $f : X \to Y$ between metric spaces is an isometry if it preserves distances: $d_X(x,y) = d_Y(f(x),f(y))$ for all $x,y \in X$ . If one exists, $X$ and $Y$ are isometric. This is the metric analogue of a homeomorphism, and it is strictly stronger: isometric $\Rightarrow$ homeomorphic, but not conversely. (Indeed, two homeomorphic spaces need not even be Hausdorff, so no metric need exist.)

Worked Example

Claim. The taxicab metric $d_T$ and the Euclidean metric $d$ on $\mathbb{R}^2$ are not isometric — even though they generate the same topology.

Suppose, for contradiction, $f : (\mathbb{R}^2, d_T) \to (\mathbb{R}^2, d)$ is an isometry. Take four points $A, B, C, D$ arranged on a unit "diamond" so that, under $d_T$ , $A$ – $B$ – $C$ – $D$ are unit steps and the diagonals span two steps. The isometry must reproduce all of these as Euclidean distances:

\begin{aligned} d_T(A,C) &= 2 &&\Rightarrow\ d\bigl(f(A),f(C)\bigr) = 2, \\ d_T(A,B) = d_T(B,C) &= 1 &&\Rightarrow\ d\bigl(f(B),f(A)\bigr)=d\bigl(f(B),f(C)\bigr)=1, \\ d_T(A,D) = d_T(C,D) &= 1 &&\Rightarrow\ d\bigl(f(D),f(A)\bigr)=d\bigl(f(D),f(C)\bigr)=1. \end{aligned}

In the Euclidean plane the only point exactly $1$ from both $f(A)$ and $f(C)$ (which are $2$ apart) is the midpoint of the segment $\overline{f(A)f(C)}$ . So $f(B)$ and $f(D)$ are both forced onto that single midpoint, giving

\begin{aligned} d\bigl(f(B),f(D)\bigr) &= 0, \\ \text{but } d_T(B,D) &= 2 \ \Rightarrow\ d\bigl(f(B),f(D)\bigr) = 2. \end{aligned}

A distance cannot be both $0$ and $2$ — contradiction. Hence no isometry exists: the two metric spaces are homeomorphic but not isometric.

Summary

Every metric space is Hausdorff; therefore a non-Hausdorff topology is not metrizable.
Continuity between metric spaces has an $\epsilon$ – $\delta$ characterization: every $\epsilon$ -ball around $f(x)$ contains the image of some $\delta$ -ball around $x$ — generalizing the calculus definition to arbitrary metrics.
One topology is finer than another exactly when its balls fit inside the coarser one's; the Euclidean, taxicab, and max metrics on $\mathbb{R}^2$ all give the same topology.
The distance between sets is the infimum of point distances; it can vanish for disjoint sets and is not a metric on subsets.
A metric is bounded if all distances stay below some $\mu$ ; $d'(x,y)=\min\{d(x,y),1\}$ is always a bounded metric inducing the same topology as $d$ .
An isometry is a distance-preserving bijection; being isometric is strictly stronger than being homeomorphic, and the Euclidean and taxicab metrics on $\mathbb{R}^2$ are homeomorphic but not isometric.