Metric Space

A set X with a metric d taking pairs of elements to real numbers such that
(i) d(x,y) >= 0 and = o only when x=y;
(ii) d(x,y) = d(y,x)
(iii) d(x,z) <= d(x,y) + d(y,z).

Examples:
(1) The Euclidean metric on Rn d(x,y) = |x-y|;
(2) The discrete metric on any x: d(x,y) = 0 if x=y otherwise d(x,y) = 1;
(3) The Hamming metric on Xn: d(x,y) = number of components in with x and y differ.

The open balls B(x,r) = set of points at distance less than r from x form the basis for the metric topology on X.

A topology is metrizable if it is the metric topology for some metric.

Compact

A topological space is compact if every cover by open sets has a finite subcover. Some writers require that a compact space also be Hausdorff (T2) but this is not usual.


Compact

A topological space is compact if every cover by open sets has a finite subcover. Some writers require that a compact space also be Hausdorff (T2) but this is not usual.


Hausdorff Space

A topological space is Hausdorff if any two distinct points have disjoint open neighbourhoods. This is also called the T2 condition.


Completeness

A Cauchy sequence in a metric space (X,d) is a sequence (xn) such that for all e>0 there exists and N = N(e) such that for all i,j >= N, d(xi,xj) < e.
A metric space is complete if every Cauchy sequence converges.

Examples:
(1) the reals with the usual metric form a complete metic space;
(2) the rationals with the usual metric do not.

A completion of a metric space (X,d) is a metric space (Y,f) which contains X as a dense subspace.
Every metric space has a completion, and all such completions are isometric, so we may speak of "the" completion.

Completeness is a metric but not a topological property. For example, the open interval (0,1) in R is not complete but is homeomorphic to the real line, which is.

The reals are the completion of the rationals.


Connectedness

A topological space is connected if it has no nontrivial partition into open sets: equivalently, there is no continuous map onto a discrete space of more than one element.
The components of a space are the maximal connected subspaces.

A space is path-connected if for any two points x,y there is a continuous map f from the unit interval [0,1] with the usual metric topology such that f(0)=x and f(1) = y.
The path components of a space are the maximal path-connected subspaces.

A space is arc-connected if for any two points x,y there is a continuous map one-to-one f from an interval [a,b in R with the usual metric topology such that f(a)=x and f(b) = y. (Note that this handles the case x=y.)

The arc components of a space are the maximal arc-connected subspaces.

Since an interval in the real line is connected (the Intermediate Values Theorem),
arc-connected => path-connected => connected.

Each implication is strict.