A Corson Compact Space is Countable if the Complement of its Diagonal is Functionally Countable

A space X is called functionally countable if ƒ (X) is countable for any continuous function ƒ : X → Ø. Given an infinite cardinal k, we prove that a compact scattered space K with d(K) > k must have a convergent k+-sequence. This result implies that a Corson compact space K is countable if the space (K × K) \ ΔK is functionally countable; here ΔK = {(x, x): x ϵ K} is the diagonal of K. We also establish that, under Jensen’s Axiom ♦, there exists a compact hereditarily separable non-metrizable compact space X such that (X × X) \ ΔX is functionally countable and show in ZFC that there exists a non-separable σ-compact space X such that (X × X) \ ΔX is functionally countable.

Structural and Chemical Controllers of the North and Northwest of Torud Based on Involved Fluid Studies, Structural and Geochemical Analyses

Chah Mura mining area in Semnan province is located 30 km southwest of Shahroud and 20 km north of Torud village with an area of 35 km2 and includes a part of 1:250,000 Torud plate. Structurally, this area is located in the northeastern part of Central Iran and in the center of the volcanic-intrusive arc of Torud-Chah Shirin. Rock units of the area are volcanic and pyroclastic, depending on the Eocene age. Exposed assemblages in the Chah Mura area, based on field and laboratory studies, can be divided into basalt, andesite, andesite-basalt, trachyandesite, trachyandesibasalt and small outcrops of pyroclastic units in the form of agglomerates and sediments of sandstone and conglomerate. Volcanic rocks are influenced by sub-volcanic masses younger than Eocene with an intermediate to basic composition, and their predominant textures are granular, porphyroid with microcrystalline to microintragranular background. Finally, the units are cut by dikes. In this area, mineralization is mainly in the control of sub-faults and subvolcanic massifs. Mineralization is in the form of vein-veinlet, filling empty and scattered space in the oxidation-supergen stage. Mineral sequences include pyrite, chalcopyrite, chalcocite, digenite and covellite, cuprite, tenorite, natural copper, malachite, azurite, and iron oxides and hydroxides. Geochemical studies indicate that copper does not correlate well with any of the base metals and depositing elements. Copper shows only a relative correlation with silver. Micrometric studies of fluid inclusions in samples from this area indicate dilution as a result of mixing hydrothermal solutions with atmospheric fluids in formation of this reserve.

STRONG COMPLETENESS OF PROVABILITY LOGIC FOR ORDINAL SPACES

Given a scattered space $\mathfrak{X} = \left( {X,\tau } \right)$ and an ordinal λ, we define a topology $\tau _{ + \lambda } $ in such a way that τ+0 = τ and, when $\mathfrak{X}$ is an ordinal with the initial segment topology, the resulting sequence {τ+λ}λ∈Ord coincides with the family of topologies $\left\{ {\mathcal{I}_\lambda } \right\}_{\lambda \in Ord} $ used by Icard, Joosten, and the second author to provide semantics for polymodal provability logics.We prove that given any scattered space $\mathfrak{X}$ of large-enough rank and any ordinal λ > 0, GL is strongly complete for τ+λ. The special case where $\mathfrak{X} = \omega ^\omega + 1$ and λ = 1 yields a strengthening of a theorem of Abashidze and Blass.

Scattered Spaces, Compactifications and an Application to Image Classification Problem

In this paper, we characterize spaces such that their one-point compactification is a scattered space. We study spaces such that their Wallman compactification is scattered and we give necessary and sufficient conditions on the subspace A to get its closure scattered. An application of scattered spaces to image classification problem is also presented.

Sequential Order under PFA

It is shown that it follows from PFA that there is no compact scattered space of height greater than ω in which the sequential order and the scattering heights coincide.

Compact semilattices with open principal filters

A locally compact semilattice with open principal filters is a zero-dimensional scattered space. Cardinal invariants of locally compact and compact semilattices with open principal filters are investigated. Structure of topological semilattices on the one-point Alexandroff compactification of an uncountable discrete space and linearly ordered compact semilattices with open principal filters are researched.

On Katětov Spaces

Recent work by Krystock, Porter, and Vermeer has emphasized the importance of the concepts of Katětov spaces and H-sets in the theory of H-closed spaces. These properties are closely related to being the θ-closure of some set and being the adherence of an open filter. This relationship is developed by establishing, among other facts, that an H-closed space in which every closed set is the θ-closure of some set is compact and the θ-closure of a subset of an H-closed space is Katětov and characterizing the open filter adhérences of a space as precisely those sets which are the image of a closed set of the absolute of the space. Also, examples are given of a countable, scattered space which is not Katětov and an H-closed space with an H-closed subspace which is not the θ-closure of any subset of the given space.

Semi-embeddings of Banach spaces which are hereditarily c0

Following Lotz, Peck and Porta [9], a continuous linear operator from one Banach space into another is called a semi-embedding if it is one-to-one and maps the closed unit ball of the domain onto a closed (hence complete) set. (Below we shall allow the codomain to be an F-space, i.e., a complete metrisable topological vector space.) One of the main results established in [9] is that if X is a compact scattered space, then every semi-embedding of C(X) into another Banach space is an isomorphism ([9], Main Theorem, (a)⇒(b)).