**Induced Uniformity**

One way to construct a uniform structure on a topological space *X* is to take the initial uniformity on *X* induced by *C*(*X*), the family of real-valued continuous functions on *X*. This is the coarsest uniformity on *X* for which all such functions are uniformly continuous. A subbase for this uniformity is given by the set of all entourages

where *f* ∈ *C*(*X*) and ε > 0.

The uniform topology generated by the above uniformity is the initial topology induced by the family *C*(*X*). In general, this topology will be coarser than the given topology on *X*. The two topologies will coincide if and only if *X* is completely regular.

Read more about this topic: Uniformizable Space

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