# Famous Theorems of Mathematics/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

such that

*d*(*x*,*y*) ≥ 0 (*non-negativity*)*d*(*x*,*y*) = 0 if and only if*x*=*y*(*identity of indiscernibles*)*d*(*x*,*y*) =*d*(*y*,*x*) (*symmetry*)*d*(*x*,*z*) ≤*d*(*x*,*y*) +*d*(*y*,*z*) (*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[edit]

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 consisting of all points q such that d(p,q) < r. The number r is called the*radius*of . If the metric space is (here the metric is assumed to be the Euclidean metric) then is known as the*open ball*with*center*p and radius r. The*closed ball*is defined for d(p,q) r. - A point p is a
*limit point*of the set E if every neighbourhood of p contains a point qp such that q E. - If p 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 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 X such that d(p,q) < M for all p 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[edit]

1. *Every neighborhood is an open set*

*Proof*: Consider a neighborhood N = . Now if q N then as d(p,q) < r we have h = r - d(p,q) > 0. Consider s . Now d(p,s) d(p,q) + d(q,s) < r - h + h = r, and so 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 contains no point q of E such that q 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 . We need to show that x . Now every neighborhood of x contains a point of so that x is not an interior point of E. Since E is open it means x E and so x . So is closed.

- Now suppose is closed. Choose x E. Then x , and so x is not a limit point of . 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.