Topology/Lesson 3

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Metric Space[edit | edit source]

A metric space has very many useful properties and they are very good for beginning topology students to study because of their relative intuitive nature.

Definition[edit | edit source]

Let be a set. Then a metric on the set is a function such that

  1. and equality holds if and only if
  2. (i.e., is symmetric).
  3. (the triangle inequality).

We can define a topology on the set using the metric as follows. For and define Then take the collection as a basis. Call this topology

If is a topological space, then we say that it is metrizable if there is a metric such that If is metrizable and is such a metric then we call the pair a metric space.

Examples[edit | edit source]

  1. The most obvious example is the space together with the metric which is called the standard metric on
  2. Any set has a metric on it. As an example, let be any set and let be given by It is left as an exercise to check that this is a metric, called the discrete metric because the topology it gives is the discrete topology.

Exercises[edit | edit source]

  1. Show that the discrete metric is a metric and that it gives the discrete topology on

Isometries[edit | edit source]

The "isomorphism" in the category of topological spaces is homeomorphism, as we saw in Lesson 2. The "isomorphism" in the category of metric spaces is called "global isometry" and is much more strict than homeomorphism.

Definition[edit | edit source]

Let and be metric spaces. Then a function is called an isometry if for all we have (i.e. if distances are preserved). If is also surjective, then we say that is a global isometry.

Examples[edit | edit source]

  1. If is any metric space, then the identity map is an isometry.
  2. If is a metric space and then the inclusion map is an isometry.

Exercises[edit | edit source]

  1. Show that every isometry is injective and therefore every global isometry is bijective.
  2. Show that every isometry is an embedding and therefore every global isometry is a homeomorphism.
  3. Find a function that is a homeomorphism but not an isometry.