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


 * $$d : M \times M \rightarrow \mathbb{R}$$

such that


 * 1) d(x, y) ≥ 0     (non-negativity)
 * 2) d(x, y) = 0   if and only if   x = y     (identity of indiscernibles)
 * 3) d(x, y) = d(y, x)      (symmetry)
 * 4) 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
Let X be a metric space. All points and sets are elements and subsets of X.


 * 1) A neighborhood of a point p is a set $$N_r(p)$$ consisting of all points q such that d(p,q) < r. The number r is called the radius of $$N_r(p)$$. If the metric space is $$R^k$$ (here the metric is assumed to be the Euclidean metric) then $$N_r(p)$$ is known as the open ball with center p and radius r. The closed ball is defined for d(p,q) $$\le$$ r.
 * 2) A point p is a limit point of the set E if every neighbourhood of p contains a point q$$\ne$$p such that q $$\in$$ E.
 * 3) If p $$\in$$ E and p is not a limit point of E then p is called an isolated point of E.
 * 4) E is closed if every limit point of E is a point of E.
 * 5) A point p is an interior point of E if there is a neighborhood N of p such that N $$\subset$$ E.
 * 6) E is open if every point of E is an interior point of E.
 * 7) E is perfect if E is closed and if every point of E is a limit point of E.
 * 8) E is bounded if there is a real number M and a point q $$\in$$ X such that d(p,q) < M for all p $$\in$$ E.
 * 9) 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 = $$N_r(p)$$. Now if q $$\in$$ N then as d(p,q) < r we have h = r - d(p,q) > 0. Consider s $$\in$$ $$N_h(q)$$. Now d(p,s) $$\le$$ d(p,q) + d(q,s) < r - h + h = r, and so $$N_h(q)$$ $$\subset$$ 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 $$N_r(p)$$ contains no point q of E such that q $$\ne$$ 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 $$E^c$$. We need to show that x $$\in E^c$$. Now every neighborhood of x contains a point of $$E^c$$ so that x is not an interior point of E. Since E is open it means x $$\notin$$ E and so x $$\in E^c$$. So $$E^c$$ is closed.


 * Now suppose $$E^c$$ is closed. Choose x $$\in$$ E. Then x $$\notin E^c$$, and so x is not a limit point of $$E^c$$. 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.