Definitions A topology on a set X is a collection U of subsets, called open sets, such that X ∈ U, the union of any subfamily of U belongs to U, and the intersection of two elements of U also belongs to U. A topology can be defined by prescribing a set V of subsets of X to be a ‘subbase’ of open sets: then define U to consist of arbitrary unions of finite intersections of elements of V. A set W is a base of open sets if every open set is a union of elements of W.

A subset F of X is closed if its complement X \ F is open. If A is any subset of X (in particular, if A is a point) a subset V of X is a neighbourhood of A if there is an open set U with A ⊆ U ⊆ V.

If Y ⊂ X is a subset of a space X with a topology U, the subspace topology on Y is given by taking as open sets the U ∩ Y with U ∈ U.

A topology is said to be Hausdorff if for any we can find U1, U2 ∈ U with x1 ∈ U1, x2 ∈ U2 and U1, U2 disjoint, i.e. U1 ∩U2 = ∅. This is a rather weak condition, and all spaces we will consider are Hausdorff. In a Hausdorff space, each point is a closed set. There are also stricter separation conditions (which hold for smoothmanifolds): a topology is completely regular if any point x and closed set F not containing it are contained in disjoint open sets, and normal if disjoint closed sets F1, F2 are contained in disjoint open sets U1, U2 ∈ U.

A mapping f : X →Y between two topological spaces is continuous if whenever V is open in Y, f−1(V) is open in X. It is a homeomorphism if f is bijective and both f and f−1 are continuous. We call f an embedding if it is injective and gives a homeomorphism between X and f (X) with the subspace topology.

An important condition on a topology is the existence of a countable base of open sets. This holds for Rn since we can take the balls with rational radii and centres having rational coordinates.