Metric space

This is a variation of topological space. View other variations of topological space

Definition

A metric space is a set $X$ along with a distance function $d:X\times X\to \mathbb {R}$ such that the following hold for all $x,y,z\in X$:

• Non-negativity: $d(x,y)\geq 0$
• Identity of indiscernibles: $d(x,x)=0\iff x=0$
• Symmetry: $d(x,y)=d(y,x)$
• Triangle inequality: $d(x,y)+d(y,z)\geq d(x,z)$

A distance function satisfying all the above three conditions is termed a metric.

A metric space is typically denoted by the ordered pair of the set and the metric, so the metric space above is $(X,d)$. However, when the metric is implicitly understood or has been pre-specified, we can omit it and simply say that $X$ is a metric space.

Induced topology

Further information: Metric induces topology

There is a natural induced topology on any metric space: the topology whose basis is open balls of positive radii about points in the metric space. Here, by open ball of radius $r$ about $x$ we mean the set of points $y$ such that $d(x,y)<r$. The fact that this works requires the use of the axioms of symmetry and triangle inequality.

A topological space which arises via the induced topology on a metric space, is termed metrizable. There may be many different metrics yielding the same topology, for instance the taxicab metric and the Euclidean metric for Euclidean space.

Category structure

With continuous maps

Further information: Category of metric spaces with continuous maps

The collection of metric spaces can be viewed as a category, with a morphism of metric spaces defined as a continuous map. A map between metric spaces is continuous if the following $\epsilon -\delta$ definition is satisfied. A map $f:(X,d)\to (X',d')$ is continuous if:

$\ \forall \ x\in X,\ \epsilon >0,\ \exists \ \delta >0,d(y,x)<\delta \ \implies \ d'(f(y),f(x))<\epsilon$

Note that under this definition, the isomorphisms do not necessarily preserve the metric -- they simply preserve the underlying topological structure.

With short maps

Further information: Category of metric spaces with short maps

The collection of metric spaces can be viewed as a category, with a morphism of metric spaces defined as a short map. A short map is a map between metric spaces that does not increase distance. A map $f:(X,d)\to (X',d')$ is termed short, or non-expanding, if:

$\ \forall x,y\in X,d(f(x),f(y))\leq d(x,y)$

Facts

• The metric is a jointly continuous function from the metric space to $\mathbb {R}$. This follows from the various axioms for the function. For full proof, refer: Metric is jointly continuous
• The map that sends a metric space to its associated topological space, is a functor. In other words, a continuous map between metric spaces is a continuous map of the underlying topological spaces. For full proof, refer: Induced topology from metric is functorial
• The topology arising from the metric induced on a subspace (by restricting the metric from the whole space) is the same as the subspace topology arising from the whole space. For full proof, refer: topology from subspace metric equals subspace topology