### Approach merotopological spaces and their completion.

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We study Čech complete and strongly Čech complete topological spaces, as well as extensions of topological spaces having these properties. Since these two types of completeness are defined by means of covering properties, it is quite natural that they should have a convenient formulation in the setting of nearness spaces and that in that setting these formulations should lead to new insights and results. Our objective here is to give an internal characterization of (and to study) those nearness...

We conduct an investigation of the relationships which exist between various generalizations of complete regularity in the setting of merotopic spaces, with particular attention to filter spaces such as Cauchy spaces and convergence spaces. Our primary contribution consists in the presentation of several counterexamples establishing the divergence of various such generalizations of complete regularity. We give examples of: (1) a contigual zero space which is not weakly regular and is not a Cauchy...

For mappings $f\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}S\to S$, where $S$ is a merotopic space equipped with a diameter function, we introduce and examine an entropy, called the $\delta $-entropy. The topological entropy and the entropy of self-mappings of metric spaces are shown to be special cases of the $\delta $-entropy. Some connections with other characteristics of self-mappings are considered. We also introduce and examine an entropy for subsets of ${S}^{N}$, which is closely connected with the $\delta $-entropy of $f\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}S\to S$.

∗ This work was partially supported by the National Foundation for Scientific Researches at the Bulgarian Ministry of Education and Science under contract no. MM-427/94.In this paper the notion of SR-proximity is introduced and in virtue of it some new proximity-type descriptions of the ordered sets of all (up to equivalence) regular, resp. completely regular, resp. locally compact extensions of a topological space are obtained. New proofs of the Smirnov Compactification Theorem [31] and of the...

We introduce the structure of a nearness on a $\sigma $-frame and construct the coreflection of the category $\mathbf{N}\sigma Frm$ of nearness $\sigma $-frames to the category $\mathbf{K}Reg\sigma Frm$ of compact regular $\sigma $-frames. This description of the Samuel compactification of a nearness $\sigma $-frame is in analogy to the construction by Baboolal and Ori for nearness frames in [1] and that of Walters for uniform $\sigma $-frames in [11]. We also construct the uniform coreflection of a nearness $\sigma $-frame, that is, the coreflection of the category of $\mathbf{N}\sigma Frm$ to the category...

Let $X$ be a completely regular Hausdorff space and, as usual, let $C\left(X\right)$ denote the ring of real-valued continuous functions on $X$. The lattice of $z$-ideals of $C\left(X\right)$ has been shown by Martínez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) $\beta X$ precisely when $X$ is a $P$-space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a $d$-ideal if whenever two elements have the same annihilator and...