Metric Space

A metric space is a set for which a distance function can be defined.

Not to be confused with a measure space.

Description

A metric space is the double (X,d) composed of:

• a given space X

• a distance or metric function d: X × X -> [0,∞)

The distance function must satisfy these conditions: for any points a, b, and c in X...

• d(a,b) ≥ 0, and d(a,b) = 0 only if a = b

• symmetry: d(a,b) = d(b,a)

• triangle inequality: d(a,b) + d(b,c) ≥ d(a,c)

Compactness

A set is compact when every cover has a finite subcover. In a metric space like R, it is always possible to design an infinite series that converges to the limit point of a set without including that limit point. Crucially however, no finite subset of these can converge to that limit point. Therefore whenever a set does not include its own limit points, there is an infinite cover with no finite subcover.

Thus, in a metric space...

• all compact sets are closed.
• given a compact set, all closed subsets are also compact.

In Rⁿ specifically, per the Heine-Borel theorem, a set being closed and bounded implies compactness, and vice versa.

Comparison to Topological Spaces

Topological spaces are a similar concept that use 'closeness' rather than a distance function. There are however some advantages to using a metric space.

Within a metric space, a set is open if every point within the set can be perturbed in any direction and remain within the set. Clearly this is only ever not the case if a point is a limit point of a set, so an open set does not include any of its limit points while a closed set includes all of them.

The topological definition of openness relies on a neighborhood topology function, i.e. for a set A the neighborhood of point p is given by N_A(p). A similar function can be defined for metric spaces; a ball containing all points less than r distance away from point p is the neighborhood N_r(p). This has a distinct advantage in that neighborhoods are always open. This is proven by demonstrating that for every point in a neighborhood, the neighborhood of that new point is a subset of the original neighborhood.

• Consider the aforementioned neighborhood of p ∈ A, i.e. N_r(p)

• Choose any point q ∈ N_r(p)

• Let a = d(p,q)

  • Note that d(p,q) = a < r

• Let r' = r - a

• Define the neighborhood of q as N_r'(q)

• Choose any point s ∈ N_r'(q)

  • Note that d(q,s) < r' = r - a

• Via the triangle inequality, d(p,q) + d(q,s) ≥ d(p,s)

• Since d(q,s) < r - a and d(p,q) = a, it follows that d(p,q) + d(q,s) < (r - a) + a = r

• Altogether d(p,s) < r, proving that N_r'(q) ⊆ N_r(p)

CategoryRicottone