**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

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