Extensions of uniform structures

Continuous uniformities

“…Some interesting facts in this line have already been shown (see [4]). Some interesting facts in this line have already been shown (see [4]).…”

“…If S is a subset of a topological space (X, 5-) then for the followino conditions, (1) is equivalent to (2) and (3) is equivalent (4). Our first theorem shows that these extensions can be taken to be prerealcompact in the case of a prerealcompact uniformly embedded subset and likewise the extension can be taken to be precompact in the case of the precompact uniformly embedded subsets.…”

“…Trivially (1) is implied by (2) and (3) is implied by (4). Trivially (1) is implied by (2) and (3) is implied by (4).…”

“…To prove sufficiency, it is necessary to recall that S is P-embedded in X if and only if °//o(S ) = qlo(X)jS x S (see [4,Theorem 7.5]) and that for any subspace S of X, q/o(X)lS ×S is always contained in ~o(S). To prove sufficiency, it is necessary to recall that S is P-embedded in X if and only if °//o(S ) = qlo(X)jS x S (see [4,Theorem 7.5]) and that for any subspace S of X, q/o(X)lS ×S is always contained in ~o(S).…”

“…In dosing we remark that for infinite cardinal numbers ~ one can define the concepts of P~-embedding (see [9]) and ~-uniformly-embedding using the uniformities q/r (see [4] and [3]). Here a 7-uniformly embedded subset would be one for which every admissible uniformity on S generated by a collection of y-separable continuous pseudometrics on S has a continuous extension.…”