-separation axioms on frames
The algebra of pseudotopologies
The arbitrary intersection of a family of open subsets with the -s-locally finite property is -semi-open.
The closures of arithmetic progressions in the common division topology on the set of positive integers
In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive integers with the base consisting of arithmetic progressions {an + b} such that if the prime number p is a factor of a, then it is also a factor of b. The topology T is called the common division topology.
The common division topology on
A topological space is totally Brown if for each and every nonempty open subsets of we have . Totally Brown spaces are connected. In this paper we consider a topology on the set of natural numbers. We then present properties of the topological space , some of them involve the closure of a set with respect to this topology, while others describe subsets which are either totally Brown or totally separated. Our theorems generalize results proved by P. Szczuka in 2013, 2014, 2016 and by...
The convergence space of minimal usco mappings
A convergence structure generalizing the order convergence structure on the set of Hausdorff continuous interval functions is defined on the set of minimal usco maps. The properties of the obtained convergence space are investigated and essential links with the pointwise convergence and the order convergence are revealed. The convergence structure can be extended to a uniform convergence structure so that the convergence space is complete. The important issue of the denseness of the subset of all...
The decomposition series of a convergence space
The essential cover and the absolute cover of a schematic space
A theorem of Gleason states that every compact space admits a projective cover. More generally, in the category of topological spaces with continuous maps, covers exist with respect to the full subcategory of extremally disconnected spaces. Such a cover of a space is called its absolute. We prove that the absolute exists within the category of schematic spaces, i.e. the spaces underlying a scheme. For a schematic space, we use the absolute to generalize Bourbaki's concept of irreducible component,...
The frame of fibrewise closed nuclei
The hyperspace of lower semicontinuity and the first power of a topological space
The Lattice of all Systems of -Ideals in a Set
The lattice of pseudotopologies on S
The notion of closedness in topological categories
In [1], various generalizations of the separation properties, the notion of closed and strongly closed points and subobjects of an object in an arbitrary topological category are given. In this paper, the relationship between various generalized separation properties as well as relationship between our separation properties and the known ones ([4], [5], [7], [9], [10], [14], [16]) are determined. Furthermore, the relationships between the notion of closedness and strongly closedness are investigated...
The quasi topology associated with a countably subadditive set function
This is a general study of an increasing, countably subadditive set function, called a capacity, and defined on the subsets of a topological space . The principal aim is the study of the “quasi-topological” properties of subsets of , or of numerical functions on , with respect to such a capacity . Analogues are obtained to various important properties of the fine topology in potential theory, notably the quasi Lindelöf principle (Doob), the existence of a fine support (Getoor), and the theorem...
Tietze Extension Theorem for n-dimensional Spaces
In this article we prove the Tietze extension theorem for an arbitrary convex compact subset of εn with a non-empty interior. This theorem states that, if T is a normal topological space, X is a closed subset of T, and A is a convex compact subset of εn with a non-empty interior, then a continuous function f : X → A can be extended to a continuous function g : T → εn. Additionally we show that a subset A is replaceable by an arbitrary subset of a topological space that is homeomorphic with a convex...
Topological characterization of the small cardinal
We show that the small cardinal number is a maximal independent family} has the following topological characterization: has a dense irresolvable countable subspace}, where denotes the Cantor cube of weight . As a consequence of this result, we have that the Cantor cube of weight has a dense countable submaximal subspace, if we assume (ZFC plus ), or if we work in the Bell-Kunen model, where and .
Topological quasitopos hulls of categories containing topological and metric objects
Topological Space Objects in a Topos II: ...-Completeness and ...-Cocompleteness.
Topological spaces with a selected subset-cardinal invariants and inequalities
Cardinal functions for topological spaces in which a subset is selected in a certain way are defined and studied. Most of the main cardinal inequalities are generalized for such spaces.
Topological Structures On Classes I