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.

Functorial polar functions

2011 ◽  
Vol 61 (3) ◽  
Author(s):  
Ricardo Carrera
Keyword(s):  
Hausdorff Spaces ◽  
Weak Unit ◽  
Closed Sets ◽  
Polar Function ◽  
Uncountable Cardinal ◽  
Functorial Polar Functions ◽  
Skeletal Maps ◽  
Polar Functions ◽  
Regular Closed

AbstractW∞ denotes the category of archimedean ℓ-groups with designated weak unit and complete ℓ-homomorphisms that preserve the weak unit. CmpT2,∞ denotes the category of compact Hausdorff spaces with continuous skeletal maps. This work introduces the concept of a functorial polar function on W∞ and its dual a functorial covering function on CmpT2,∞.We demonstrate that functorial polar functions give rise to reflective hull classes in W ∞ and that functorial covering functions give rise to coreflective covering classes in CmpT 2,∞. We generate a variety of reflective and coreflecitve subcategories and prove that for any regular uncountable cardinal α, the class of α-projectable ℓ-groups is reflective in W ∞, and the class of α-disconnected compact Hausdorff spaces is coreflective in CmpT 2,∞. Lastly, the notion of a functorial polar function (resp. functorial covering function) is generalized to sublattices of polars (resp. sublattices of regular closed sets).

scholarly journals On coabsolute paracompact spaces

1979 ◽  
Vol 27 (2) ◽  
pp. 248-256 ◽  
Author(s):  
Catherine L. Gates
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Algebra Isomorphism ◽  
Closed Sets ◽  
Local Finiteness ◽  
Paracompact Spaces ◽  
Regular Closed

AbstractWe are interested in determining whether two spaces are coabsolute by comparing their Boolean algebras of regular closed sets. It is known that when the spaces are compact Hausdorff they are coabsolute precisely when the Boolean algebras of regular closed sets are isomorphic; but in general this condition is not strong enough to insure that the spaces be coabsolute. In this paper we show that for paracompact Hausdorff spaces, the spaces are coabsolute when the Boolean algebra isomorphism and its inverse ‘preserve’ local finiteness, and for locally compact paracompact Hausdorff spaces, the spaces are coabsolute when the collections of compact regular closed subsets are ‘isomorphic’.

Quasihomeomorphisms and Skula spaces

2021 ◽  
Author(s):  
Othman Echi
Keyword(s):  
Topological Space ◽  
Alternative Proof ◽  
Manuscripta Math ◽  
Alexandroff Space ◽  
Closed Sets ◽  
Locally Closed Sets ◽  
Reflection Formula

Let [Formula: see text] be a topological space. By the Skula topology (or the [Formula: see text]-topology) on [Formula: see text], we mean the topology [Formula: see text] on [Formula: see text] with basis the collection of all [Formula: see text]-locally closed sets of [Formula: see text], the resulting space [Formula: see text] will be denoted by [Formula: see text]. We show that the following results hold: (1) [Formula: see text] is an Alexandroff space if and only if the [Formula: see text]-reflection [Formula: see text] of [Formula: see text] is a [Formula: see text]-space. (2) [Formula: see text] is a Noetherian space if and only if [Formula: see text] is finite. (3) If we denote by [Formula: see text] the Alexandroff extension of [Formula: see text], then [Formula: see text] if and only if [Formula: see text] is a Noetherian quasisober space. We also give an alternative proof of a result due to Simmons concerning the iterated Skula spaces, namely, [Formula: see text]. A space is said to be clopen if its open sets are also closed. In [R. E. Hoffmann, Irreducible filters and sober spaces, Manuscripta Math. 22 (1977) 365–380], Hoffmann introduced a refinement clopen topology [Formula: see text] of [Formula: see text]: The indiscrete components of [Formula: see text] are of the form [Formula: see text], where [Formula: see text] and [Formula: see text] is the intersection of all open sets of [Formula: see text] containing [Formula: see text] (equivalently, [Formula: see text]). We show that [Formula: see text]

scholarly journals On generalized γµ-closed sets and related continuity

2021 ◽  
Vol 13 (2) ◽  
pp. 483-493
Author(s):  
Ritu Sen
Keyword(s):  
Topological Space ◽  
Main Interest ◽  
Closed Sets ◽  
Separation Axioms ◽  
Preservation Theorems ◽  
Generalized Topological Space ◽  
New Type ◽  
Weak Separation

Abstract In this paper our main interest is to introduce a new type of generalized open sets defined in terms of an operation on a generalized topological space. We have studied some properties of this newly defined sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have studied some preservation theorems in terms of some irresolute functions.

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.

A Measure Theoretical Version of the Aleksandrov Theorem

2008 ◽  
Vol 15 (1) ◽  
pp. 53-61
Author(s):  
Majid Gazor
Keyword(s):  
Borel Measure ◽  
Measure Theory ◽  
Ordinal Number ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Closed Sets

Abstract In this paper a theorem analogous to the Aleksandrov theorem is presented in terms of measure theory. Furthermore, we introduce the condensation rank of Hausdorff spaces and prove that any ordinal number is associated with the condensation rank of an appropriate locally compact totally imperfect space. This space is equipped with a probability Borel measure which is outer regular, vanishes at singletons, and is also inner regular in the sense of closed sets.

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].

On Product of Radón Measures

1969 ◽  
Vol 12 (4) ◽  
pp. 427-444 ◽  
Author(s):  
M. C. Godfrey ◽  
M. Sion
Keyword(s):  
Direct Reference ◽  
Outer Measure ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Radon Measures ◽  
Closed Sets

Let X, Y be locally compact Hausdorff spaces and μ, ν be Radón outer measures on X and Y respectively. The classical product outer measure ϕ on X × Y generated by measurable rectangles, without direct reference to the topology, turns out to have some serious drawbacks. For example, one can only prove that closed sets (and hence Baire sets) are ϕ-measurable. It is unknown, even when X and Y are compact, whether closed sets are ϕ-measurable.

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