On the question of representation of ultrafilters and their application in extension constructions

On the structure of ultrafilters and properties related to convergence in topological spaces

“…Let us consider this issue in more detail, following the approach of [11,14,21] and using the notation of the preceding section. Since, in particular, τ ∈ π[X], we have [13, definition (5.3)] the set is fulfilled.…”

“…This provides a sufficiently complete answer to the question of whether (8.2) is feasible in the class of continuous mappings from (X, τ ) to (Y, θ). Another approach to the construction of mappings from F lim [X; L; Y ; θ], where L ∈ Π[X], was implemented in [11,14,21] for the case when (Y, θ) is a Tychonoff power of a topological space that could be equipped with a complete metric. The role of such mappings was played by operators with tier components.…”

“…3.6]) in the procedure from [7] are not implemented constructively, which complicates their practical application (actually, it is these ultrafilters that are responsible for the realization of nontrivial instances of asymptotic behavior). The representations established in [9] were developed further in [11][12][13][14]. They were, however, oriented towards the application of measurable spaces in the traditional sense (algebras and semialgebras of sets were used).…”

“…In the present paper, we focus on this case. More exactly, we study the following question, which is important from the viewpoint of the constructions from [11,13,14]: when does a continuous mapping from a topological space to a topological space map "open" (made of open sets) ultrafilters of the first space to converging filter bases of the second space?…”

Products of Ultrafilters and Maximal Linked Systems on Widely Understood Measurable Spaces

“…In what follows, we consider questions related to a representation of ultrafilters on x∈X L x as products (7.7). In this connection, we recall [4]. But, in the present constructions, we use a scheme based on (4.6).…”