Disconnection in the Alexandroff duplicate

<p>It was demonstrated in [2] that the Alexandroff duplicate of the Čech-Stone compactification of the naturals is not extremally disconnected. The question was raised as to whether the Alexandroff duplicate of a non-discrete extremally disconnected space can ever be extremally disconnected. We answer this question in the affirmative; an example of van Douwen is significant. In a slightly different direction we also characterize when the Alexandroff duplicate of a space is a P-space as well as when it is an almost P-space.</p>

A study of m-N-extremally Disconnected Spaces With Respect to, Maximum mX-N-open Sets

The aim of this research is to prove the idea of maximum mX-N-open set, m-N-extremally disconnected with respect to t and provide some definitions by utilizing the idea of mX-N-open sets. Some properties of these sets are studied.

Star-Hurewicz Modulo an Ideal Property In Topological Spaces

In this paper, a class of star-Hurewicz modulo an ideal spaces is introduced and studied. For an ideal <em>K</em> of finite subsets of N, a characterization of weakly star-<em>K</em>-Hurewicz extremally disconnected spaces is given using ideal. It is shown that star-Hurewicz modulo an ideal property is hereditary under clopen subspaces. In this manner we obtained relationships of star-Hurewicz modulo an ideal property with other existing Hurewicz properties in literature.

Two extensions of the stone duality to the category of zero-dimensional Hausdorff spaces

Extending the Stone Duality Theorem, we prove two duality theorems for the category ZHaus of zero-dimensional Hausdorff spaces and continuous maps. They extend also the Tarski Duality Theorem; the latter is even derived from one of them. We prove as well two new duality theorems for the category EDTych of extremally disconnected Tychonoff spaces and continuous maps. Also, we describe two categories which are dually equivalent to the category ZComp of zero-dimensional Hausdorff compactifications of zero-dimensional Hausdorff spaces and obtain as a corollary the Dwinger Theorem about zero-dimensional compactifications of a zero-dimensional Hausdorff space.

£-Single Valued Extremally Disconnected Ideal Neutrosophic Topological Spaces

This paper aims to mark out new concepts of r-single valued neutrosophic sets, called r-single valued neutrosophic £-closed and £-open sets. The definition of £-single valued neutrosophic irresolute mapping is provided and its characteristic properties are discussed. Moreover, the concepts of £-single valued neutrosophic extremally disconnected and £-single valued neutrosophic normal spaces are established. As a result, a useful implication diagram between the r-single valued neutrosophic ideal open sets is obtained. Finally, some kinds of separation axioms, namely r-single valued neutrosophic ideal-Ri (r-SVNIRi, for short), where i={0,1,2,3}, and r-single valued neutrosophic ideal-Tj (r-SVNITj, for short), where j={1,2,212,3,4}, are introduced. Some of their characterizations, fundamental properties, and the relations between these notions have been studied.

How to have more things by forgetting how to count them

Cohen’s first model is a model of Zermelo–Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from this Dedekind-finite set to some infinite ordinal κ . In the case that we force the function to be injective, it turns out that the resulting model is the same as adding κ Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekind-finite set. In the case where the function is merely surjective it turns out that we do not add any reals, sets of ordinals, or collapse any Dedekind-finite sets. This motivates the question if there is any combinatorial condition on a Dedekind-finite set A which characterises when a forcing will preserve its Dedekind-finiteness or not add new sets of ordinals. We answer this question in the case of ‘Adding a Cohen subset’ by presenting a varied list of conditions each equivalent to the preservation of Dedekind-finiteness. For example, 2 A is extremally disconnected, or [ A ] < ω is Dedekind-finite.

(Λ, Sp)-closed Sets and Related Topics in Topological Spaces

The purpose of the present paper is to introduce the concepts of Λsp-sets, (Λ, sp)-open sets and (Λ, sp)- closed sets which are defined by utilizing the notions of β-open sets and β-closed sets. Some characterizations of Λsp-submaximal spaces, Λsp-extremally disconnected spaces and Λsp-hyperconnected spaces are established. Moreover, several characterizations of upper and lower (Λ, sp)-continuous multifunctions are investigated.