It is known that for Ω a connected separable metric space, a function Ω → R which at every point attains a weak local extremummust have countable image. Let Ω ⊂ R be a bounded domain and f : Ω → C a measurable, bounded, quasi-continuous function. For n ∈ Z+, we show that if |f | has countable image then f attains weak local extremum at each point of a dense open subset. We show by examples the ...

Countable Lawvere theories model computational effects such as exceptions, side-effects, interactive input/output, nondeterminism and probabilistic nondeterminism. The category of countable Lawvere theories has sums, tensors, and distributive tensors, modelling natural combinations of such effects. It is also closed under taking images. Enrichment in a category such as Cpo allows one to extend ...

In this paper, the notions of countable S∗-compactness is introduced in L-topological spaces based on the notion of S∗-compactness. An S∗-compact L-set is countably S∗-compact. If L = [0, 1], then countable strong compactness implies countable S∗-compactness and countable S∗-compactness implies countable F-compactness, but each inverse is not true. The intersection of a countably S∗-compact L-s...

Let $ { mathcal{R}} L$ be the ring of real-valued continuous functions on a frame $L$ as the pointfree  version of $C(X)$, the ring of all real-valued continuous functions on a topological space $X$. Since $C_c(X)$ is the largest subring of $C(X)$ whose elements have countable image, this motivates us to present the pointfree  version of $C_c(X).$The main aim of this paper is to present t...

Z. Li characterized spaces with a weak-development consisting of point-countable sfp-covers by pseudo-sequence-covering, quotient, and π-s-image of metric spaces. In this paper, we omit ”pseudo-sequence-covering” in the above result, and prove that a space has a weak-development consisting of point-countable sfp-covers iff it is a quotient, and π-s-image of a metric space.

in an earlier work we showed that for ordered fields f not isomorphic to the reals r, there are continuous 1-1 unctions on [0, 1]f which map some interior point to a boundary point of the image (and so are not open). here we show that over closed bounded intervals in the rationals q as well as in all non-archimedean ordered fields of countable cofinality, there are uniformly continuous 1-1 func...

We employ a forcing approach to extending Boolean algebras. A link between some forcings and some cardinal functions on Boolean algebras is found and exploited. We find the following applications: 1) We make Fedorchuk’s method more flexible, obtaining, for every cardinal λ of uncountable cofinality, a consistent example of a Boolean algebra Aλ whose every infinite homomorphic image is of cardin...

Consider a measure-preserving action Γ y (X,μ) of a countable group Γ and a measurable cocycle α : X × Γ → Aut(Y ) with countable image, where (X,μ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ y X is non-amenable, then there does not exist a countable-to-one Borel homomorphism from the orbit equivalence...

Let I be an ideal on N. A mapping f : X ? Y is called I-covering provided a sequence {yn}n?N I-converging to point y in Y, there {xn}n?N converging x such that ?1(y) and each xn ?1(yn). In this paper we study the spaces with certain I-cs-networks investigate characterization of images metric under mappings, which prompts us discover I-cs -networks. The following main results are obtained: (1) s...

We show that PFA implies that every perfect pre–image of ω1 of countable tightness contains a closed copy of ω1.