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.
Read more about this topic: Uniform Space
Famous quotes containing the words uniform and/or spaces:
“When a uniform exercise of kindness to prisoners on our part has been returned by as uniform severity on the part of our enemies, you must excuse me for saying it is high time, by other lessons, to teach respect to the dictates of humanity; in such a case, retaliation becomes an act of benevolence.”
—Thomas Jefferson (1743–1826)
“Le silence éternel de ces espaces infinis m’effraie. The eternal silence of these infinite spaces frightens me.”
—Blaise Pascal (1623–1662)