topology induced by uniform structure

Let 𝒰 be a uniform structure on a set X. We define a subset A to be open if and only if for each xA there exists an entourage U𝒰 such that whenever (x,y)U, then yA.

Let us verify that this defines a topologyMathworldPlanetmathPlanetmath on X.

Clearly, the subsets and X are open. If A and B are two open sets, then for each xAB, there exist an entourage U such that, whenever (x,y)U, then yA, and an entourage V such that, whenever (x,y)V, then yB. Consider the entourage UV: whenever (x,y)UV, then yAB, hence AB is open.

Suppose is an arbitrary family of open subsets. For each x, there exists A such that xA. Let U be the entourage whose existence is granted by the definition of open set. We have that whenever (x,y)U, then yA; hence y, which concludes the proof.

Title topology induced by uniform structure
Canonical name TopologyInducedByUniformStructure
Date of creation 2013-03-22 12:46:44
Last modified on 2013-03-22 12:46:44
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 7
Author Mathprof (13753)
Entry type Derivation
Classification msc 54E15
Related topic UniformNeighborhood
Defines uniform topology