Strong Cauchy completeness in uniform frames

On preparacompactness of uniform frames

“…Filter characterizations of Lindelöfness of a regular frame in terms of clustering σ-filters (given in [15]) follows, noting that a subset T of L is conservative if for each S T and each…”

“…F is a weakly Cauchy filter in case sec F meets every uniform cover and (L, μ) is then strongly Cauchy complete if every weakly Cauchy filter clusters. Every strongly Cauchy complete uniform frame is Cauchy complete (see [15]). (1) If F is a Cauchy (respectively, weakly Cauchy) filter on L, then h * (F ) is a Cauchy (respectively, weakly Cauchy) filter on M .…”

“…We recall from [15] that a uniform frame (L, μ) is preparacompact if whenever F is a weakly Cauchy filter on L there is a Cauchy filter G on L such that F G. [15] shows that if (L, μ) is preparacompact then so is (M, ν).…”

“…Thus the completion (CL, Cμ) of a uniform frame (L, μ) is always paracompact. That the completion (CL, Cμ) is compact if and only if the uniform frame (L, μ) is precompact is well-known (see [5] or [15]). …”

“…In [15] the pointfree notion of preparacompactness was introduced as a filter generalization of the uniform property of precompactness. We continue with this notion from [15] to set out and characterize those uniform frames with Lindelöf and compact completions via preparacompactess of the uniformity on a completely regular frame L induced by the completion map γ L : CL → L. To achieve this endeavour, we first require the concept of strongly Cauchy complete introduced for uniform frames in [16] and further investigated in [15]. This notion on the fine uniformity of a completely regular frame will be shown to be equivalent to paracompactness.…”