The arbitrary intersection of a family of open subsets with the -s-locally finite property is -semi-open.
The characterizations of upper approximation operators based on special coverings
In this paper, we discuss the approximation operators [...] apr¯NS and [...] apr¯S which are based on NS(U) and S. We not only obtain some properties of NS(U) and S, but also give examples to show some special properties. We also study sufficient and necessary conditions when they become closure operators. In addition, we give general and topological characterizations of the covering for two types of covering-based upper approximation operators being closure operators.
The family of -density type topologies
We investigate a family of topologies introduced similarly as the -density topology. In particular, we compare these topologies with respect to inclusion and we look for conditions under which these topologies are identical.
The investigation of some topologies on finite sets.
The Lattice of all Systems of -Ideals in a Set
The lattice of precompact topologies on .
The lattice of topologies of topological -groups
The topology of the unit interval is not uniquely determined by its continuous self maps among set systems
Tightness and resolvability
We prove resolvability and maximal resolvability of topological spaces having countable tightness with some additional properties. For this purpose, we introduce some new versions of countable tightness. We also construct a couple of examples of irresolvable spaces.
Topologies generated by closed intervals.
Topologies generated by ideals
A topological space is said to be generated by an ideal if for all and all there is in such that , and is said to be weakly generated by if whenever a subset of contains for every with , then itself is closed. An important class of examples are the so called weakly discretely generated spaces (which include sequential, scattered and compact Hausdorff spaces). Another paradigmatic example is the class of Alexandroff spaces which corresponds to spaces generated by finite sets....
Topology on ordered fields
An ordered field is a field which has a linear order and the order topology by this order. For a subfield of an ordered field, we give characterizations for to be Dedekind-complete or Archimedean in terms of the order topology and the subspace topology on .
Transitivity and partial order
In this paper we find a one-to-one correspondence between transitive relations and partial orders. On the basis of this correspondence we deduce the recurrence formula for enumeration of their numbers. We also determine the number of all transitive relations on an arbitrary -element set up to .