Definition:Topology Induced by Metric

Definition

Let $M = \left({A, d}\right)$ be a metric space.

Definition 1

The topology on the metric space $M = \struct {A, d}$ induced by (the metric) $d$ is defined as the set $\tau$ of all open sets of $M$.

Definition 2

The topology on the metric space $M = \struct {A, d}$ induced by (the metric) $d$ is defined as the topology $\tau$ generated by the basis consisting of the set of all open $\epsilon$-balls in $M$.

Also known as

The topological space which is so induced is also known as the topological space associated with the (given) metric space.

When the context is clear, the phase metric topology for $d$ can be used.

The metric space whose metric induces this topology can be said to give rise to the topological space.

Also see

• Equivalence of Definitions of Topology Induced by Metric

• Metric Induces Topology, in which it is shown that $\tau$ is, in fact, a topology on $M$
• Definition:Subspace Topology (sometimes also called the induced topology)