Metric Spaces

Metric & Metric Space

A metric on a set is a map such that

1. Positivity: with equality if and only if ;
2. Symmetry: for every ;
3. Triangle inequality: for every .

We call a metric space.

e.g.

• The discrete metric on any non-empty set is defined by setting and if ;
• Any norm on a vector space gives rise to a metric on by setting . For example, on , we have metrics for given by which are induced from $L^{p}$ norms.
• Let . Then the uniform metric on is given by More generally, over a set of functions , the uniform metric is defined as
• Consider the set of integers. Let be a prime. For any with , there is a unique such that , where is not divisible by . The following is called the -adic metric:
• Let and be two metric spaces. Then for any , we have defines a metric on .
• Let be the set of all words (finite sequences of 26 symbols). Then the Levenshtein (spelling) distance between word and is the minimum number of ‘edits’ required to change from to , where an ‘edit’ is any one of (i) insertion of a symbol (ii) deletion of a symbol (iii) change of a letter.
• Let be a graph. The combinatorial metric defined on the vertices of is the minimal number of edges required to join the two vertices. (For this definition, we need to assume that each pair of vertices can be joined by a path in the graph).
  

Metric Subspace

Metric Subspace

Let be a metric space and be a subset of . Define Then is a metric on which is called the induced metric on and is called a metric subspace of .

Proposition

Suppose is a nonempty metric subspace of . Given . Let denote the open ball in of radius with center . Then

Proof For any , and , so . Conversely, for any , and , so , hence .

Proposition

Let be a metric space and a subspace. Let , then

• is open in iff with open in .
• is closed in iff with closed in .

e.g. Consider in . and are open in , but is not open in . and are closed in , but is not closed in .