we show that the category of convergence approach spaces is a simultaneously reective and coreective subcategory of the category of latticevalued limit spaces. further we study the preservation of diagonal conditions, which characterize approach spaces. it is shown that the category of preapproach spaces is a simultaneously reective and coreective subcategory of the category of lattice-valued p...

the concept of ${mathscr{f}}_{st}$-fundamentality is introduced in uniform spaces, generated by some filter ${mathscr{f}}$. its equivalence to the concept of ${mathscr{f}}$-convergence in uniform spaces is proved. this convergence generalizes many kinds of convergence, including the well-known statistical convergence.

We apply Preuss' concept of $mbbe$-connectedness to the categories of lattice-valued uniform convergence spaces and of lattice-valued uniform spaces. A space is uniformly $mbbe$-connected if the only uniformly continuous mappings from the space to a space in the class $mbbe$ are the constant mappings. We develop the basic theory for $mbbe$-connected sets, including the product theorem. Furtherm...

We develop a general framework for various lattice-valued, probabilistic and approach uniform convergence spaces. To this end, we use the concept of $s$-stratified $LM$-filter, where $L$ and $M$ are suitable frames. A stratified $LMN$-uniform convergence tower is then a family of structures indexed by a quantale $N$. For different choices of $L,M$ and $N$ we obtain the lattice-valued, probabili...

We show that the category of convergence approach spaces is a simultaneously reective and coreective subcategory of the category of latticevalued limit spaces. Further we study the preservation of diagonal conditions, which characterize approach spaces. It is shown that the category of preapproach spaces is a simultaneously reective and coreective subcategory of the category of lattice-valued p...

The aim of this paper is to set up appropriate uniform convergence spaces in which to reformulate and enrich the Order Completion Method [25] for nonlinear PDEs. In this regard, we consider an appropriate space ML (X) of normal lower semi-continuous functions. The space ML (X) appears in the ring theory of C (X) and its various extensions [15], as well as in the theory of nonlinear, PDEs [25] a...

This paper focuses on the relationships between stratified $L$-conver-gence spaces, stratified strong $L$-convergence spaces and stratifiedlevelwise  $L$-convergence spaces. It has been known that: (1) astratified $L$-convergence space is precisely a left-continuousstratified levelwise $L$-convergence space; and (2) a  stratifiedstrong $L$-convergence space is naturally a stratified $L$-converg...