Uniform Space - Topology of Uniform Spaces

Topology of Uniform Spaces

Every uniform space X becomes a topological space by defining a subset O of X to be open if and only if for every x in O there exists an entourage V such that V is a subset of O. In this topology, the neighbourhood filter of a point x is {V : V∈Φ}. This can be proved with a recursive use of the existence of a "half-size" entourage. Compared to a general topological space the existence of the uniform structure makes possible the comparison of sizes of neighbourhoods: V and V are considered to be of the "same size".

The topology defined by a uniform structure is said to be induced by the uniformity. A uniform structure on a topological space is compatible with the topology if the topology defined by the uniform structure coincides with the original topology. In general several different uniform structures can be compatible with a given topology on X.