Mathematics

Let E be a metric space. We introduce a notion of connectedness index of E, which is the Hausdor? dimension of the union of non-trivial connected components of E. We show that the connectedness index of a fractal cube E is strictly less than the Hausdor? dimension of E provided that E possesses a trivial connected component. Hence the connectedness index is a new Lipschitz invariant. Moreover, we investigate the relation between the connectedness index and topological Hausdor? dimension.

In this paper, we provide a direct approach to K -reflections of T 0 spaces. For a full subcategory K of the category of all T 0 spaces and a T 0 space X , let K(X)={A⊆X:A is closed and for any continuous mapping f:X⟶Y to a K -space Y , there exists a unique y A ∈Y such that f(A) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ = { y A } ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ } and P H (K(X)) the space of K(X) endowed with the lower Vietoris topology. It is proved that if P H (K(X)) is a K -space, then the pair ⟨ X k = P H (K(X)), η X ⟩ , where η X :X⟶ X k , x↦ {x} ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ , is the K -reflection of X . We call K an adequate category if for any T 0 space X , P H (K(X)) is a K -space. Therefore, if K is adequate, then K is reflective in Top 0 . It is shown that the category of all sober spaces, that of all d -spaces, that of all well-filtered spaces and the Keimel and Lawson's category are all adequate, and hence are all reflective in Top 0 . Some major properties of K -spaces and K -reflections of T 0 spaces are investigated. In particular, it is proved that if K is adequate, then the K -reflection preserves finite products of T 0 spaces. Our study also leads to a number of problems, whose answering will deepen our understanding of the related spaces and their categorical structures.

Functional representations of the capacity monad based on the max and min operations were considered in \cite{Ra1} and \cite{Ny1}. Nykyforchyn considered in \cite{Ny2} some alternative monad structure for the possibility capacity functor based on the max and usual multiplication operations. We show that such capacity monad (which we call the capacity multiplication monad) has a functional representation, i.e. the space of capacities on a compactum X can be naturally embedded (with preserving of the monad structure) in some space of functionals on C(X,I) . We also describe this space of functionals in terms of properties of functionals.

We prove that if X is a paracompact space, Y is a metric space and f:X→Y is a functionally fragmented map, then (i) f is σ -discrete and functionally F σ -measurable; (ii) f is a Baire-one function, if Y is weak adhesive and weak locally adhesive for X ; (iii) f is countably functionally fragmented, if X is Lindelöff. This result generalizes one theorem of Rene Baire on classification of barely continuous functions.

Let \mathscr{F}=(F_n) be a sequence of nonempty finite subsets of \omega such that \lim_n |F_n|=\infty and define the ideal \mathcal{I}(\mathscr{F}):=\left\{A\subseteq \omega: |A\cap F_n|/|F_n|\to 0~\mbox{as}~n\to \infty \right\}. The case F_n=\{1,\ldots,n\} corresponds to the classical case of density zero ideal. We show that \mathcal{I}(\mathscr{F}) is an analytic P-ideal but not F_{\sigma} . As a consequence, we show that the set of real bounded sequences which are \mathcal{I}(\mathscr{F}) -convergent to 0 is not complemented in \ell_\infty .

With a simple generic approach, we develop a classification that encodes and measures the strength of completeness (or compactness) properties in various types of spaces and ordered structures. The approach also allows us to encode notions of functions being contractive in these spaces and structures. As a sample of possible applications we discuss metric spaces, ultrametric spaces, ordered groups and fields, topological spaces, partially ordered sets, and lattices. We describe several notions of completeness in these spaces and structures and determine their respective strengths. In order to illustrate some consequences of the levels of strength, we give examples of generic fixed point theorems which then can be specialized to theorems in various applications which work with contracting functions and some completeness property of the underlying space. Ball spaces are nonempty sets of nonempty subsets of a given set. They are called spherically complete if every chain of balls has a nonempty intersection. This is all that is needed for the encoding of completeness notions. We discuss operations on the sets of balls to determine when they lead to larger sets of balls; if so, then the properties of the so obtained new ball spaces are determined. The operations can lead to increased level of strength, or to ball spaces of newly constructed structures, such as products. Further, the general framework makes it possible to transfer concepts and approaches from one application to the other; as examples we discuss theorems analogous to the Knaster--Tarski Fixed Point Theorem for lattices and theorems analogous to the Tychonoff Theorem for topological spaces.

We prove that the existence of a Borel lower density operator (a Borel lifting) with respect to the σ -ideal of countable sets, for an uncountable Polish space, is equivalent to the Continuum Hypothesis.

We build an example of a metric space with transfinite asymptotic dimension 2ω .

We construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension 2ω+1 .

We construct a metrizable Lawson semitopological semilattice X whose partial order ≤ X ={(x,y)∈X×X:xy=x} is not closed in X×X . This resolves a problem posed earlier by the authors.