THE MODAL LOGIC OF STONE SPACES: DIAMOND AS DERIVATIVE

2010 ◽  
Vol 3 (1) ◽  
pp. 26-40 ◽  
Author(s):  
GURAM BEZHANISHVILI ◽  
LEO ESAKIA ◽  
DAVID GABELAIA
Keyword(s):  
Modal Logic ◽  
Topological Space ◽  
Hausdorff Spaces ◽  
Derived Set ◽  
Stone Spaces

We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces is K4 and the modal logic of weakly scattered Stone spaces is K4G. As a corollary, we obtain that K4 is also the modal logic of compact Hausdorff spaces and K4G is the modal logic of weakly scattered compact Hausdorff spaces.

Epimorphisms From S(X) onto S(Y)

1986 ◽  
Vol 38 (3) ◽  
pp. 538-551 ◽  
Author(s):  
K. D. Magill ◽  
P. R. Misra ◽  
U. B. Tewari
Keyword(s):  
Topological Space ◽  
Hausdorff Spaces ◽  
Severe Restriction ◽  
Completely Regular ◽  
Image Position

1. Introduction. In this paper, the expression topological space will always mean generated space, that is any T1 space X for whichforms a subbasis for the closed subsets of X. This is not at all a severe restriction since generated spaces include all completely regular Hausdorff spaces which contain an arc as well as all 0-dimensional Hausdorff spaces [3, pp. 198-201], [4].The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. This paper really grew out of our efforts to determine all those congruences σ on S(X) such that S(X)/σ is isomorphic to S(Y) for some space Y.

Local Topological Properties of Maps and Open Extensions of Maps

1977 ◽  
Vol 29 (6) ◽  
pp. 1121-1128
Author(s):  
J. K. Kohli
Keyword(s):  
Topological Space ◽  
Open Subset ◽  
Countable Union ◽  
Dense Open Subset ◽  
Topological Properties ◽  
Hausdorff Spaces ◽  
Local Homeomorphism ◽  
Open Map

A σ-discrete set in a topological space is a set which is a countable union of discrete closed subsets. A mapping ƒ : X ⟶ Y from a topological space X into a topological space Y is said to be σ-discrete (countable) if each fibre ƒ-1(y), y ϵ Y is σ-discrete (countable). In 1936, Alexandroff showed that every open map of a bounded multiplicity between Hausdorff spaces is a local homeomorphism on a dense open subset of the domain [2].

A note on λ-compact spaces

2013 ◽  
Vol 63 (6) ◽  
Author(s):  
O. Karamzadeh ◽  
M. Namdari ◽  
M. Siavoshi
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Compact Spaces

AbstractWe extend the well-known and important fact that “a topological space X is compact if and only if every ideal in C(X) is fixed”, to more general topological spaces. Some interesting consequences are also observed. In particular, the maximality of compact Hausdorff spaces with respect to the property of compactness is generalized and the topological spaces with this generalized property are characterized.

scholarly journals Minimal TUD spaces

2002 ◽  
Vol 3 (1) ◽  
pp. 55 ◽  
Author(s):  
A.E. McCluskey ◽  
W.S. Watson
Keyword(s):  
Topological Space ◽  
Partial Order ◽  
Linear Order ◽  
Closed Sets ◽  
Separation Axioms ◽  
Derived Set ◽  
Weak Separation

<p>A topological space is T<sub>UD</sub> if the derived set of each point is the union of disjoint closed sets. We show that there is a minimal T<sub>UD</sub> space which is not just the Alexandroff topology on a linear order. Indeed the structure of the underlying partial order of a minimal T<sub>UD</sub> space can be quite complex. This contrasts sharply with the known results on minimality for weak separation axioms.</p>

Types and Stone spaces

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Topological Space ◽  
Model Theory ◽  
Algebraic Structure ◽  
Possible Worlds ◽  
Elementary Extension ◽  
Philosophical Significance ◽  
Related Algebra ◽  
Contemporary Study ◽  
Stone Spaces

Types are one of the cornerstones of contemporary model theory. Simply put, a type is the collection of formulas satisfied by an element of some elementary extension. The types can be organised in an algebraic structure known as a Lindenbaum algebra. But the contemporary study of types also treats them as the points of a certain kind of topological space. These spaces, called ‘Stone spaces’, illustrate the richness of moving back-and-forth between algebraic and topological perspectives. Further, one of the most central notions of contemporary model theory—namely stability—is simply a constraint on the cardinality of these spaces. We close the chapter by discussing a related algebra-topology ‘duality’ from metaphysics, concerning whether to treat propositions as sets of possible worlds or vice-versa. We show that suitable regimentations of these two rival metaphysical approaches are biinterpretable (in the sense of chapter 5), and discuss the philosophical significance of this rapprochement.

scholarly journals Compact and extremally disconnected spaces

2004 ◽  
Vol 2004 (20) ◽  
pp. 1047-1056
Author(s):  
Bhamini M. P. Nayar
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Closed Subset ◽  
Hausdorff Space ◽  
Hausdorff Spaces ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Extremally Disconnected

Viglino defined a Hausdorff topological space to beC-compact if each closed subset of the space is anH-set in the sense of Veličko. In this paper, we study the class of Hausdorff spaces characterized by the property that each closed subset is anS-set in the sense of Dickman and Krystock. Such spaces are calledC-s-compact. Recently, the notion of strongly subclosed relation, introduced by Joseph, has been utilized to characterizeC-compact spaces as those with the property that each function from the space to a Hausdorff space with a strongly subclosed inverse is closed. Here, it is shown thatC-s-compact spaces are characterized by the property that each function from the space to a Hausdorff space with a strongly sub-semiclosed inverse is a closed function. It is established that this class of spaces is the same as the class of Hausdorff, compact, and extremally disconnected spaces. The class ofC-s-compact spaces is properly contained in the class ofC-compact spaces as well as in the class ofS-closed spaces of Thompson. In general, a compact space need not beC-s-compact. The product of twoC-s-compact spaces need not beC-s-compact.

scholarly journals Perfect Mappings and Spaces of Countable Type

1970 ◽  
Vol 22 (6) ◽  
pp. 1208-1210 ◽  
Author(s):  
J. E. Vaughan
Keyword(s):  
Topological Space ◽  
Affirmative Answer ◽  
Regular Space ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Countable Type ◽  
Open Set

In [1, p. 41, Theorem 3.10] Arhangel'skiï proved that the perfect image of a completely regular space of countable type is of countable type, and he asked [1, p. 60, problem 4] if a similar result held for regular or Hausdorff spaces. In this paper, it is proved that the perfect image of a space of countable type is of countable type, provided that the image is Hausdorff or regular. An affirmative answer to both of Arhangel'skiï's questions follows immediately from this. Arhangel'skiï made use of the Stone-Čech compactification in the proof of his result, but the proofs below are of a different nature.Let X be a topological space and let K ⊂ X. A collection of open sets is called a base at K provided that for every open set W ⊃ K there exists such that K ⊂ U ⊂ W. Clearly, we may assume that every member of contains K.

scholarly journals Reduced coproducts of compact Hausdorff spaces

1987 ◽  
Vol 52 (2) ◽  
pp. 404-424 ◽  
Author(s):  
Paul Bankston
Keyword(s):  
Topological Structure ◽  
Boolean Algebras ◽  
Stone Space ◽  
Hausdorff Spaces ◽  
Topological Construction ◽  
Stone Spaces

AbstractBy analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the “reduced coproduct”, which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the “ultracoproduct” can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces.

scholarly journals Maximal perfect spaces

1972 ◽  
Vol 7 (3) ◽  
pp. 429-436
Author(s):  
Ivan Baggs
Keyword(s):  
Topological Space ◽  
Limit Point ◽  
Hausdorff Space ◽  
Hausdorff Spaces

Let (X, T) be a topological space (we assume T1. throughout) where every point is a limit point. The purpose of this note is to present an internal construction of a maximal perfect topology on (X, T). The existence of a maximal connected Hausdorff space has not been demonstrated. However, this construction of a maximal perfect topology is useful in constructing connected Hausdorff spaces which cannot be embedded in a maximal connected Hausdorff space.

scholarly journals Characterizations of s-closed Hausdorff spaces

1991 ◽  
Vol 51 (2) ◽  
pp. 300-304
Author(s):  
Takashi Noiri
Keyword(s):  
Topological Space ◽  
Hausdorff Spaces ◽  
Closed Sets ◽  
Regular Closed

A topological space X is said to be S-closed if every cover of X by regular closed sets of X has a finite subcover. In this note some characterizations of S-closed Hausdorff spaces are obtained.

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