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) = |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 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 |

## Connectedness
A topological space is |