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:
“We call ourselves a free nation, and yet we let ourselves be told what cabs we can and can’t take by a man at a hotel door, simply because he has a drum major’s uniform on.”
—Robert Benchley (1889–1945)
“When I consider the short duration of my life, swallowed up in the eternity before and after, the little space which I fill and even can see, engulfed in the infinite immensity of spaces of which I am ignorant and which know me not, I am frightened and am astonished at being here rather than there. For there is no reason why here rather than there, why now rather than then.”
—Blaise Pascal (1623–1662)