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 R^{n} d(x,y) = xy;
(2) The discrete metric on any x: d(x,y) = 0 if x=y otherwise d(x,y) = 1;
(3) The Hamming metric on X^{n}: 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.
CompactA 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. 
CompactA 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 SpaceA 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 (x_{n}) such that for all e>0 there exists and N = N(e) such that for all i,j >= N, d(x_{i},x_{j}) < e.

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.
