Metric Space

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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

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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

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  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

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

Isometries

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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

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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

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  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

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  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.