The Timpson Lyceum · General Topology (Point Set Topology)

Metric Spaces – Lecture 16 (Topology)

Metric Spaces – Lecture 16

Overview

This lecture closes out our work on homeomorphisms and then pivots to a major new structure: the metric. We first nail down embeddings (and the special cases of arcs/paths and simple closed curves), then prove that the Hausdorff property is preserved by homeomorphism. From there we reset and introduce metric spaces — sets equipped with a genuine notion of distance, not just proximity. We define a metric by its three defining properties, prove that absolute value is a metric on $\mathbb{R}$, survey several metrics on $\mathbb{R}^2$ (Euclidean, taxicab, max), and show how every metric induces a topology — the metric topology — so that every metric space is automatically a topological space.

Key Concepts

Embedding. A function $f : X \to Y$ between topological spaces is an embedding if it maps $X$ homeomorphically onto its image $f(X)$, where $f(X)$ carries the subspace topology from $Y$. The image looks "topologically identical" to $X$, sitting inside $Y$.

• Arc / path: the image of an embedding $f : [-1,1] \to X$ of the closed interval. The interval's endpoints are included.
• Simple closed curve: the image of an embedding $g : S^1 \to X$ of the circle (only the bounding points of the circle, no interior or exterior).
• Knot: an embedding of $S^1$ into $\mathbb{R}^3$ — the same idea, one dimension up.

Hausdorff property. A topological space is Hausdorff if any two distinct points can be "housed off" from one another: for points $x \neq y$ there exist disjoint open sets $U, V$ with $x \in U$, $y \in V$, $U \cap V = \varnothing$. This is the first genuine topological property — one a space either has or lacks. Nice/intuitive spaces (anything from a metric) are Hausdorff; pathological ones (e.g. the finite-complement topology on $\mathbb{R}$, where any two nonempty open sets intersect) are not.

Hausdorff is a topological invariant. If $X \cong Y$ (homeomorphic) and $X$ is Hausdorff, then so is $Y$ — a homeomorphism preserves all topological structure, just as congruence preserves "being a right triangle."

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

• Positivity / identity: $d(x,y) \geq 0$, and $d(x,y) = 0 \iff x = y$.
• Symmetry: $d(x,y) = d(y,x)$.
• Triangle inequality: $d(x,z) \leq d(x,y) + d(y,z)$.

The pair $(X, d)$ is a metric space. Where a topology only answers "are these points close?" yes/no, a metric reports exactly how far apart any two points are — strictly more structure.

Metrics on $\mathbb{R}^2$. For $p = (p_1, p_2)$ and $q = (q_1, q_2)$:

• Euclidean (standard) metric: $d(p,q) = \sqrt{(q_1 - p_1)^2 + (q_2 - p_2)^2}$ — straight-line distance, from the Pythagorean theorem.
• Taxicab metric: $d_t(p,q) = |p_1 - q_1| + |p_2 - q_2|$ — horizontal plus vertical travel, as on a city grid.
• Max metric: $d_m(p,q) = \max\{\,|p_1 - q_1|,\ |p_2 - q_2|\,\}$ — the larger of the two coordinate displacements.

(The min of the two displacements is not a metric: two distinct points differing only in one coordinate would get distance $0$, violating the identity property.)

Open and closed balls. In a metric space $(X, d)$, for $x \in X$ and $\varepsilon > 0$:

$B(x, \varepsilon) = \{\, y \in X : d(x,y) < \varepsilon \,\}, \qquad \overline{B}(x, \varepsilon) = \{\, y \in X : d(x,y) \leq \varepsilon \,\}.$

The same balls look different under different metrics: the Euclidean ball is a disk, the taxicab ball is a diamond (rhombus), and the max ball is a square. For the same $x$ and $\varepsilon$, the taxicab diamond sits inside the Euclidean disk, which sits inside the max square.

Metric topology. Let $(X, d)$ be a metric space and let $\beta$ be the collection of all open balls. Then $\beta$ is a basis, and the topology it generates is the topology induced by $d$ (the metric topology). A set $U$ is open in this topology iff for every $y \in U$ there is some $\varepsilon > 0$ with $B(y, \varepsilon) \subseteq U$. Consequence: every metric space is automatically a topological space (the converse fails). On $\mathbb{R}^n$ the Euclidean metric induces exactly the standard topology; on $\mathbb{R}$ the open ball $B(x,\varepsilon)$ is just the open interval $(x - \varepsilon,\ x + \varepsilon)$.

Function-space metric. On $C[a,b]$, the set of continuous functions $f : [a,b] \to \mathbb{R}$, one natural metric is the area between the graphs:

$d(f, g) = \int_a^b |f(x) - g(x)|\, dx.$

Two functions are distance $0$ apart exactly when they are the same function. This treats functions themselves as the "points" of a metric space.

Worked Example

Claim. Absolute value defines a metric on $\mathbb{R}$ via $d(x,y) = |x - y|$.

We use two lemmas: $x \leq |x|$ for all $x$, and $|-x| = |x|$.

Positivity / identity. $|x - y| \geq 0$ always, and $|x - y| = 0 \iff x - y = 0 \iff x = y$.

Symmetry. Apply $|-x| = |x|$ to the difference:

Triangle inequality. First establish the general fact $|a + b| \leq |a| + |b|$ by cases on the sign of $a + b$:

Now take any $x, y, z$ and split the difference $x - z$ through $y$ — letting $a = x - y$ and $b = y - z$:

All three properties hold, so $d(x,y) = |x - y|$ is a metric on $\mathbb{R}$. (This is just the Euclidean metric on $\mathbb{R}^1$, since $\sqrt{(x - y)^2} = |x - y|$.)

Summary

• An embedding maps a space homeomorphically onto its image; an arc/path embeds $[-1,1]$, a simple closed curve embeds $S^1$, and a knot embeds $S^1$ into $\mathbb{R}^3$.
• The Hausdorff property (distinct points have disjoint open neighborhoods) is a topological invariant — preserved by homeomorphism.
• A metric $d : X \times X \to \mathbb{R}$ must satisfy positivity/identity, symmetry, and the triangle inequality; it gives exact distances, strictly more structure than a topology.
• Many metrics exist on the same set: Euclidean, taxicab, and max on $\mathbb{R}^2$ (the min fails); their open balls are disks, diamonds, and squares respectively, and even spaces of functions can be metrized.
• Every metric induces a topology (the metric topology) whose basis is the set of all open balls, so every metric space is a topological space; the Euclidean metric recovers the standard topology on $\mathbb{R}^n$.