Quantale-Valued Uniformizations of Quantale-Valued Generalizations of Approach Groups

We introduce the categories of quantale-valued approach uniform spaces and quantale-valued uniform gauge spaces, and prove that they are topological categories. We first show that the category of quantale-valued uniform gauge spaces is a full bireflective subcategory of the category of quantale-valued approach uniform spaces and; second, we prove that only under strong restrictions on the quantale these two categories are isomorphic. Besides presenting embeddings of the category of quantale-valued metric spaces into the categories of quantale-valued approach uniform spaces as well as quantale-valued uniform gauge spaces, we show that every quantale-valued approach system group and quantale-valued gauge group has a natural underlying quantale-valued approach uniform space, respectively, a quantale-valued uniform gauge space.

Categorical properties of L-fuzzifying convergence spaces

In this paper, categorical properties of L-fuzzifying convergence spaces are investigated. It is shown that (1) the category L-FYC of L-fuzzifying convergence spaces is a strong topological universe; (2) the category L-FYKC of L-fuzzifying Kent convergence spaces, as a bireflective and bicoreflective subcategory of L-FYC, is also a strong topological universe; (3) the category L-FYLC of L-fuzzifying limit spaces, as a bireflective subcategory of L-FYKC, is a topological universe.

Neighborhood spaces

Neighborhood spaces, pretopological spaces, and closure spaces are topological space generalizations which can be characterized by means of their associated interior (or closure) operators. The category NBD of neighborhood spaces and continuous maps contains PRTOP as a bicoreflective subcategory and CLS as a bireflective subcategory, whereas TOP is bireflectively embedded in PRTOP and bicoreflectively embedded in CLS. Initial and final structures are described in these categories, and it is shown that the Tychonov theorem holds in all of them. In order to describe a successful convergence theory in NBD, it is necessary to replace filters by more generalp-stacks.

Completion of a Cauchy space without theT2-restriction on the space

A completion of a Cauchy space is obtained without theT2restriction on the space. This completion enjoys the universal property as well. The class of all Cauchy spaces with a special class of morphisms calleds-maps form a subcategoryCHY' ofCHY. A completion functor is defined for this subcategory. The completion subcategory ofCHY' turns out to be a bireflective subcategory ofCHY'. This theory is applied to obtain a characterization of Cauchy spaces which allow regular completion.

Non archimedean metric induced fuzzy uniform spaces

It is shown that the category of non-Archimedean metric spaces with 1-Lipschitz maps can be embedded as a coreflective non-bireflective subcategory in the category of fuzzy uniform spaces. Consequential characterizations of topological and uniform properties are derived.