# Definition:Topology Induced by Metric

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## 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 phrase **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

- 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**)