Mathematics

In 1999, Molodtsov initiated the theory of soft sets as a new mathematical tool for dealing with uncertainties in many fields of applied sciences. In 2011, Shabir and Naz introduced and studied the notion of soft topological spaces, also defining and investigating many new soft properties as generalization of the classical ones. In this paper, we introduce the notions of soft separation between soft points and soft closed sets in order to obtain a generalization of the well-known Embedding Lemma to the class of soft topological spaces.

We prove that the locally convex space C p (X) of continuous real-valued functions on a Tychonoff space X equipped with the topology of pointwise convergence is distinguished if and only if X is a Δ -space in the sense of \cite {Knight}. As an application of this characterization theorem we obtain the following results: 1) If X is a Čech-complete (in particular, compact) space such that C p (X) is distinguished, then X is scattered. 2) For every separable compact space of the Isbell--Mrówka type X , the space C p (X) is distinguished. 3) If X is the compact space of ordinals [0, ω 1 ] , then C p (X) is not distinguished. We observe that the existence of an uncountable separable metrizable space X such that C p (X) is distinguished, is independent of ZFC. We explore also the question to which extent the class of Δ -spaces is invariant under basic topological operations.

We prove that an almost zero-dimensional space X is an Erdős space factor if and only if X has a Sierpiński stratification of C-sets. We apply this characterization to spaces which are countable unions of C-set Erdős space factors. We show that the Erdős space E is unstable by giving strongly σ -complete and nowhere σ -complete examples of almost zero-dimensional F σδ -spaces which are not Erdős space factors. This answers a question by Dijkstra and van Mill.

We investigate the notion of productive cellularity of arbitrary posets and topological spaces. Particularly, by working with families of antichains ordered with reverse inclusion, we give necessary and sufficient conditions to determine whether a poset or a topological space is productively ccc.

We characterize the uniform convergence points set of a pointwisely convergent sequence of real-valued functions defined on a perfectly normal space. We prove that if X is a perfectly normal space which can be covered by a disjoint sequence of dense subsets and A⊆X , then A is the set of points of the uniform convergence for some convergent sequence ( f n ) n∈ω of functions f n :X→R if and only if A is G δ -set which contains all isolated points of X . This result generalizes a theorem of Ján Borsík published in 2019.

We present a bound for the weak Lindelöf number of the G δ -modification of a Hausdorff space which implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: |X|≤ 2 L(X)χ(X) (Arhangel'skii) and |X|≤ 2 c(X)χ(X) (Hajnal-Juhasz). This solves a question that goes back to Bell, Ginsburg and Woods and is mentioned in Hodel's survey on Arhangel'skii's Theorem. In contrast to previous attempts we do not need any separation axiom beyond T 2 .

The inequality |X|≤ 2 χ(X) has been proved to be true for Lindelöf spaces (Arhangel'ski\uı, 1969), H -closed spaces (Dow-Porter, 1982) and ccc spaces (Hajnal-Juász 1967), by quite different arguments. We present a common extension of all these properties which allows us to give a unified proof of these three theorems.

We prove that (1) for any complete lattice L , the set D(L) of all nonempty saturated compact subsets of the Scott space of L is a complete Heyting algebra (with the reverse inclusion order); and (2) if the Scott space of a complete lattice L is non-sober, then the Scott space of D(L) is non-sober. Using these results and the Isbell's example of a non-sober complete lattice, we deduce that there is a complete Heyting algebra whose Scott space is non-sober, thus give a positive answer to a problem posed by Jung. We will also prove that a T 0 space is well-filtered iff its upper space (the set D(X) of all nonempty saturated compact subsets of X equipped with the upper Vietoris topology) is well-filtered, which answers another open problem.

We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space X , the function space C p (X) is not countable dense homogeneous. This answers a question posed recently by R. Hernández-Gutiérrez. We also conclude that, for any infinite dimensional Banach space E (dual Banach space E ∗ ), the space E equipped with the weak topology ( E ∗ with the weak ∗ topology) is not countable dense homogeneous. We generalize some results of Hrušák, Zamora Avilés, and Hernández-Gutiérrez concerning countable dense homogeneous products.

Under Martin's Axiom we construct a Boolean countably compact topological group whose square is not countably pracompact.