equivalent condition for being a fundamental system of entourages
(B1) If , , then contains an element of .
(B2) Each element of contains the diagonal .
(B3) For any , the inverse relation of contains an element of .
Suppose is a fundamental system of entourages for a uniformity . Verification of axiom (B2) is immediate, since and each entourage is already required to contain the diagonal of . We will prove that satisfies (B1); the proofs that (B3) and (B4) hold are analogous.
Let , be entourages in . Since is closed under binary intersections, . By the definition of fundamental system of entourages, since , there exists an entourage such that . Thus satisfies axioms (B1) through (B4).
To prove the converse, define a family of subsets of by
By construction, each element of contains an element of , so all that remains is to show that is a uniformity. Suppose is a subset of that contains an element . By the definition of , there exists some such that . Since , it follows that , so satisfies the requirement for membership in . Thus is closed under taking supersets. The remaining axioms for a uniformity follow directly from the appropriate axioms for the fundamental system of entourages by applying the axiom we have just checked. Hence is a fundamental system of entourages for a uniformity on . ∎
|Title||equivalent condition for being a fundamental system of entourages|
|Date of creation||2013-03-22 16:30:02|
|Last modified on||2013-03-22 16:30:02|
|Last modified by||mps (409)|