Metric Spaces
A metric for a set X is a mapping d of X × X into the nonnegative real numbers satisfying the following conditions for all x,y,z ∈ X:
| M1: | d(x,x) = 0 | |
| M2: | d(x,z) ≤ d(x, y) + d(y,z) | |
| M3: | d(x, y) = d(y,x) | |
| M4: | if x ≠ y, d(x, y) > 0. |
We call d(x, y) the distance between x and y. If d satisfies only M1, M2, and M4 it is called quasimetric, while if it satisfies M1, M2, and M3 it is called a pseudometric. It is possible to use a metric to define a topology on X by taking as a basis all open balls B(x,
) = {y ∈ X|d(x, y) < }. A topological space together with a metric giving its topology is called a metric space. Although a single metric will yield a unique topology on a given set, it is possible to find more than one metric which will yield the same topology. In fact, there are always an infinite number of metrics which will yield the same metric space (Example 134).
Every metric space is Hausdorff, since B (p,) ∩ B (q,) = ∅, if < d(p,q)/2, and also T5. For suppose A and B are separated subsets of a metric space X; then each point x ∈ A has a neighborhood B(x,∈x) disjoint from B, and each point y ∈ B has a neighborhood B(y,∈x) disjoint from A. Then 
are disjoint open neighborhoods of A and B, respectively. Thus metric spaces are completely normal, and, by a similar argument, perfectly normal. Therefore metric spaces satisfy every Ti separation property. Furthermore, every metric space is fully T4, thus fully normal and paracompact.
Much of the structure of countability and compactness is also simplified in metric spaces. Since {B(x,1/n)|n = 1, 2, 3, …} is a countable local basis at x, each metric space is first countable. If {xi} is a countable dense subset of X, the balls B(xi,1/n) form a countable base for the topology on X. So for metric spaces separability implies, and is therefore equivalent to, second countability.
The same is true of metric spaces which are Lindelöf, since in such spaces, for each integer k, the open covers {B(x,1/k)|x ∈ X} have countable sub-covers. The union of all such subcovers is a countable base for X. Thus every Lindelöf metric space is also second countable.
Since each metric space X is first countable, sequential compactness is equivalent to countable compactness, which, since X is Ti, is equivalent to weak countable compactness. More important, countable compactness in metric spaces is equivalent to compactness, since every countably compact metric space is separable: for each n, a countably compact metric space can be covered by finitely many balls B(xin,1/n), so {xin} is a countable dense subset. Thus each countably compact metric space is second countable, and every countably compact second countable space is compact.
Since metric spaces are HausdorfT, the concepts of local compactness and strong local compactness are equivalent. So in metric space, we have a much simplified implication chart (Figure 10); that these implications do not reverse is shown by the counterexamples listed in Figure 11.
Figure 10.
Although in general the metric structure of a space does not appreciably simplify its connectedness properties, we can show that every metric space which is extremally disconnected is discrete. For in any metric space, each point p can be written as the intersection of the closed metric balls 
. Then either U or the complementary set of annuli is an open set which has p as a noninterior limit point provided p is not open. So if X is not discrete, it cannot be extremally disconnected.
Figure 11.
COMPLETE METRIC SPACES
Heuristically, compactness is related to the size of a space in that it determines how many small open sets are required for a cover. In a metric space, the radius of an open ball can be used as a precise measure of the size of the small open sets. Thus we call a subset E of a metric space X totally bounded (precompact) iff for every > 0, E may be covered by a finite collection of open balls of radius . We call such a cover an -net. A subset E is called bounded if there exists a real number β such that d(x, y) ≤ β wherever x,y ∈ E; the least bound of E is called the diameter of E. Clearly every totally bounded set is bounded, but not conversely (Example 134) ; furthermore, totally bounded is not a topological property, since it is not preserved by homeomorphisms (Example 134).
Every compact metric space is totally bounded, since every covering by -balls has a finite subcover, and every totally bounded set is second countable, since the union of -nets for = 1, 
, 
, . . . forms a countable basis. But neither of these implications reverses (Examples 30.10 and 134).
To discover the reason that totally bounded sets may fail to be compact, we must examine the convergent sequences. A sequence {xn} in à metric space (X,d) is called a Cauchy sequence iff for every > 0 there exists an integer N such that d(xm,xn) < wherever m,n > N. Obviously, every convergent sequence is a Cauchy sequence; but the converse fails (Example 32.1). So we define a complete metric space as one in which every Cauchy sequence converges to some point in the space, or equivalently that the intersection of every nested sequence of closed balls with radii tending to zero is nonempty. (A sequence {En} of sets is nested iff En ⊃ En+1 for all n.) If the radii do not tend to zero, this condition need not be implied by completeness (Example 135). Now every compact metric space is complete, and more important, every complete and totally bounded metric space is compact.
We will call a topological space (X,τ) topologically complete if there exists a metric d giving the topology τ such that (X,d) is a complete metric space. Topological completeness is a topological property which is weakly hereditary, though not hereditary (Example 30). Clearly every compact metric space is complete; though the converse is not true (Example 28), it is true that a metric space is compact iff it is complete in every equivalent metric. The famous Baire category theorem states that every topologically complete metric space is second category.
A completion of a metric space X is any complete metric space which contains a dense subset to which X is isometric, that is, to which there is a bijection which is distance preserving. All metric spaces have completions and even more surprising, all of the completions of a given space are isometric. Furthermore completeness is preserved by isometries but unlike topological completeness not by homeomorphisms (Example 32.10).
METRIZABILITY
A topological space (X,τ) is called metrizable if there exists a metric d which yields the topology τ. Every regular second countable space is metrizable, but not conversely (Example 3) ; in fact, a topological space is metrizable iff it is regular and has a σ-locally finite base, that is, a base which is the countable union of locally finite families. Although this requirement is very close to paracompactness, and though every metric space is paracompact, there exist regular paracompact spaces which are nonmetrizable (Examples 51 and 141).
UNIFORMITIES
A quasiuniformity on a set X is a collection U of subsets of X × X which satisfies the following axioms:
| U1: | For all u ∈ U, Δ ⊂ u, where Δ = {(x,x)|x ∈ X}. | |
| U2: | For u ∈ U and v ∈ U, u ∩ v ∈ U. | |
| U3: | If u ∈ U and u ⊂ v ⊂ X × X then v ∈ U. | |
| U4: | For all u ∈ U, there is v ∈ U such that v ○ v ⊂ u where ○ is defined by u ○ v = {(x,z)| there is a y ∈ X such that (x,y) ∈ v and (y,z) ∈ u}. |
The quasiuniformity U is a uniformity if the following additional condition is satisfied:
| U5: | If u ∈ U, then u−1 ∈ U where u−1 = {(y,x)|(x,y) ∈ U}. |
A set u which is an element of the quasiuniformity U is called an entourage (or a relation). The entourage u is said to be symmetric if u = u-1. The set Δ is called the diagonal of X × X. The quasiuniformity U is said to be separated if the intersection of all the members of U is the diagonal Δ.
The first three axioms say that every quasiuniformity on a set X is a fiiter on X × X. Further, a quasiuniformity U is a uniformity iff there is a symmetric base for U, that is a filter base of symmetric sets.
Every quasiuniformity U on a set X yields a topology τ on X by taking as a neighborhood system for X the sets u(x) where u ∈ U and u(x) = {y|(x,y) ∈ u} there may be more than one quasiuniformity generating a given topology (Example 44). If two quasiuniformities generate the same topology on the set X, they are said to be compatible. A set X with a quasiuniformity U and the topology τ generated by U is said to be a quasiuniform space and we may use the notation ((X, U),τ) to denote this or the shorter notation (X, U) where τ is understood to be the topology generated on X by U. A topological space (X, τ) is said to be quasiuni-formizable if there is a quasiuniformity U such that ((X,U),τ) is a quasiuniform space.
The problem of when a topological space (X,τ) is quasiuniformizable or uniformizable is simpler than the corresponding metrization problem. If (X,τ) is a topological space, the set U = {uG|uG = (G × G) ∪ (X - G) × X and G 
τ } is a filter subbase for a quasiuniformity on X which generates τ, and thus every topological space is quasiuniformizable. A topological space (X,τ) is uniformizable iff it is a T3 space.
METRIC UNIFORMITIES
If (X,d) is a pseudometric space, then the family U of all sets u which contain a set of the form u = {〈x,y〉|d(x, y) < } is a uniformity on X, which yields the same topology as the pseudometric d. Such a uniformity is called pseudometrizable (or, if appropriate, metrizable). Not every uniformity which yields a metrizable topological space need be metrizable (Example 44).