View source for The Book of Mathematical Proofs/Analysis/Metric Spaces

A '''metric space''' is a tuple (''M'',''d'') where ''M'' is a set and ''d'' is a metric on ''M'', that is, a function 

:<math>d : M \times M \rightarrow \mathbb{R}</math>

such that

# ''d''(''x'', ''y'') ≥ 0 &nbsp;&nbsp;&nbsp; (''non-negativity'')
# ''d''(''x'', ''y'') = 0 &nbsp; if and only if &nbsp; ''x'' = ''y'' &nbsp;&nbsp;&nbsp; (''identity of indiscernibles'')
# ''d''(''x'', ''y'') = ''d''(''y'', ''x'')  &nbsp;&nbsp;&nbsp; (''symmetry'')
# ''d''(''x'', ''z'') ≤ ''d''(''x'', ''y'') + ''d''(''y'', ''z'') &nbsp;&nbsp;&nbsp;  (''triangle inequality'').

The function ''d'' is also called ''distance function'' or simply ''distance''. Often ''d'' is omitted and one just writes ''M'' for a metric space if it is clear from the context what metric is used.

==Basic definitions==
Let X be a metric space. All points and sets are elements and subsets of X.

#A ''neighborhood'' of a point p is a set <math>N_r(p)</math> consisting of all points q such that d(p,q) < r. The number r is called the ''radius'' of <math>N_r(p)</math>. If the metric space is <math>R^k</math> (here the metric is assumed to be the [http://en.wikipedia.org/wiki/Euclidean_metric#Definition Euclidean metric]) then <math>N_r(p)</math> is known as the ''open ball'' with ''center'' p and radius r. The ''closed ball'' is defined for d(p,q) <math>\le</math> r. 
# A point p is a ''limit point'' of the set E if every neighbourhood of p contains a point q<math>\ne</math>p such that q <math>\in</math> E.
#If p <math>\in</math> E and p is not a limit point of E then p is called an ''isolated point'' of E.
#E is ''closed'' if every limit point of E is a point of E.
#A point p is an ''interior point'' of E if there is a neighborhood N of p such that N <math>\subset</math> E.
#E is ''open'' if every point of E is an interior point of E.
#E is ''perfect'' if E is closed and if every point of E is a limit point of E.
#E is ''bounded'' if there is a real number M and a point q <math>\in</math> X such that d(p,q) < M for all p <math>\in</math> E.
#E is ''dense'' in X every point of X is a limit point of E or a point of E (or both).

==Basic proofs==

1. ''Every neighborhood is an open set''

:''Proof'': Consider a neighborhood N = <math>N_r(p)</math>. Now if q <math>\in</math> N then as d(p,q) < r we have h =  d(p,q) - r > 0. Consider s <math>\in</math> <math>N_h(q)</math>. Now d(p,s) <math>\le</math> d(p,q) + d(q,s) < r - h + h = r, and so <math>N_h(q)</math> <math>\subset</math> N. Thus q is an interior point of N.

2. ''If p is a limit point of a set E, then every neighborhood of p contains infinitely many points of E''

:''Proof'': Suppose there is a neighborhood N of p which contains only a finite number of points of E. Let r be the minimum of the distances of these points from p. The minimum of a finite set of positive numbers is clearly positive so that r > 0. The neighborhood <math>N_r(p)</math> contains no point q of E such that q <math>\ne</math> p which contradicts the fact that p is a limit point of E.

3. ''A finite set has no limit points''   

:''Proof'': This is obvious from the proof 2.

4. ''A set is open if and only if its complement is closed.''

:''Proof'': Suppose E is open and x is a limit point of <math>E^c</math>. We need to show that x <math>\in E^c</math>. Now every neighborhood of x contains a point of <math>E^c</math> so that x is not an interior point of E. Since E is open it means x <math>\notin</math> E and so x <math>\in E^c</math>. So <math>E^c</math> is closed. 

:Now suppose <math>E^c</math> is closed. Choose x <math>\in</math> E. Then x <math>\notin E^c</math>, and so x is not a limit point of <math>E^c</math>. So there must be a neighborhood of x entirely inside E. So x is an interior point of E and so E is open. 

5. ''A set is closed if and only if its complement is open.''

:''Proof'': This is obvious from the proof 4.

[[Category:The Book of Mathematical Proofs]]
[[Category:Mathematical Proofs]]