Metric Topology

Open balls as a basis, and when a set counts as open.

The idea

The metric topology turns a set equipped with a distance function into a topological space. It is the bridge between analysis and point-set topology: through it the line, the plane, and every other space carrying a metric become topological spaces, and the epsilon arguments of analysis become statements about open sets.

The construction imitates the open interval. Given a metric $d$ on $X$, the open ball of radius $\epsilon > 0$ about a point $x$ is $B_d(x, \epsilon) = \{\, y \in X : d(x, y) < \epsilon \,\},$ the set of points strictly nearer to $x$ than $\epsilon$. These balls form a basis, and the topology they generate is the metric topology induced by $d$: a set $U$ is open exactly when every $x \in U$ admits some $\epsilon > 0$ with $B_d(x, \epsilon) \subseteq U$, where the radius may depend on the point. On $\mathbb{R}$ with $d(x, y) = |x - y|$ the balls are the intervals $(x - \epsilon, x + \epsilon)$, and the metric topology is exactly the usual topology of the line.

Different metrics can induce the same topology, and when they do, no topological statement distinguishes them: a topology records less information than a metric.

Ways to work on it

• Walkthrough. Open balls, the open-set condition, and why single points are not open.
• Practice. Fit a ball inside an open set and compute distances.
• Hardest. Show that capping distances at 1 leaves the topology unchanged.

Not sure where to start? Take the ten-question placement test.