Difference between revisions of "Metric space"
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− | + | A [[Normed space|normed space]] is a special case of a metric space, to see the relationships between metric spaces and others see: [[Subtypes of topological spaces]] | |
==Definition of a metric space== | ==Definition of a metric space== | ||
A metric space is a set <math>X</math> coupled with a "distance function"<ref name="Topology">Introduction to Topology - Bert Mendelson</ref>: | A metric space is a set <math>X</math> coupled with a "distance function"<ref name="Topology">Introduction to Topology - Bert Mendelson</ref>: |
Revision as of 22:26, 11 July 2015
A normed space is a special case of a metric space, to see the relationships between metric spaces and others see: Subtypes of topological spaces
Contents
[hide]Definition of a metric space
A metric space is a set X coupled with a "distance function"[1]:
- d:X×X→R or sometimes
- d:X×X→R+[2]
With the properties that for x,y,z∈X:
- d(x,y)≥0
- d(x,y)=0⟺x=y
- d(x,y)=d(y,x) - Symmetry
- d(x,z)≤d(x,y)+d(y,z) - the Triangle inequality
We will denote a metric space as (X,d) (as (X,d:X×X→R) is too long and Mathematicians are lazy) or simply X if it is obvious which metric we are talking about on X
Examples of metrics
Euclidian Metric
The Euclidian metric on Rn is defined as follows: For x=(x1,...,xn)∈Rn and y=(y1,...,yn)∈Rn we define the Euclidian metric by:
dEuclidian(x,y)=√n∑i=1((xi−yi)2)
Proof that this is a metric
Discreet Metric
This is a useless metric, but is a metric and induces the Discreet Topology on X, where the topology is the powerset of X, P(X).
It is given by:
- ddiscreet(x,y)={0x=y1otherwise
Note: it is sometimes called the trivial metric[3]
Proof that this is a metric