Separation axioms and frame representation of some topological facts

Pultr, Aleš; Tozzi, Anna

doi:10.1007/bf00878507

Cited by 23 publications

(5 citation statements)

References 8 publications

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“…Translated to the category of locales this proposition asserts that any closed surjective localic map preserves stratifiability. Since axiom T D is precisely equivalent with representability of closed continuous maps as closed frame homomorphisms (see [29] or [27,III.7.3.1]), we have also the following corollary from the preceding result: Corollary 7.11. Let X and Y be T D -spaces and f : X → Y a closed surjective continuous map.…”

Section: Stratifiable Framesmentioning

confidence: 84%

Monotone normality and stratifiability from a pointfree point of view

García¹,

Picado²,

Vicente³

2014

Topology and its Applications

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Monotone normality is usually defined in the class of T 1 spaces. In this paper we study it under the weaker condition of subfitness, a separation condition that originates in pointfree topology. In particular, we extend some well known characterizations of these spaces to the subfit context (notably, their hereditary property and the preservation under surjective continuous closed maps) and present a similar study for stratifiable spaces, an important subclass of monotonically normal spaces. In the second part of the paper, we extend further these ideas to the lattice theoretic setting. In particular, we give the pointfree analogues of the previous results on monotonically normal spaces and introduce and investigate the natural pointfree counterpart of stratifiable spaces.

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Section: Stratifiable Framesmentioning

confidence: 84%

Monotone normality and stratifiability from a pointfree point of view

García¹,

Picado²,

Vicente³

2014

Topology and its Applications

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“…This is an axiom stronger than T 0 and weaker than T 1 and it was introduced in Aull and Thron (1963). It has been used (for instance in Pultr and Tozzi (1994)) in order to answer the question of when a topological space can be completely recovered from its frame of opens. The Skula space of a space X, denoted as Sk(X), is the space defined as follows.…”

Section: The Axiom T Dmentioning

confidence: 99%

Revisiting the relation between subspaces and sublocales

Suarez¹

2020

Preprint

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We revisit results concerning the connection between subspaces of a space and sublocales of its locale of open sets. The approach we present is based on the notion of sublocale as a concrete subcollection of a locale. We characterize the frames L such that the spatial sublocales of S(L) perfectly represent the subspaces of pt(L). We prove choice-free, weak versions of the results by Niefield and Rosenthal characterizing those frames such that all their sublocales are spatial. We do so by using a notion of essential prime which does not rely on the existence of enough minimal primes above every element. We will re-prove Simmons' result that spaces such that the sublocales of Ω(X) perfectly represent their subspaces are exactly the scattered spaces. We will characterize scattered spaces in terms of a strong form of essentiality for primes. We apply these characterizations to show that, when L is a spatial frame and a coframe, pt(L) is scattered if and only if it is T D , and this holds if and only if all the primes of L are completely prime.

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“…This is an axiom stronger than T 0 and weaker than T 1 and it was introduced in [2]. It has been used (for instance in [11]) in order to answer the question of when a topological space can be completely recovered from its frame of opens. The Skula space of a space X, denoted as Sk(X), is the space defined as follows.…”

Section: The Axiom T Dmentioning

confidence: 99%

The coframe of D-sublocales of a locale and the $T_D$ duality

Arrieta,

Suarez

2020

Preprint

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The notion of D-sublocale is explored. This is the notion analogue to that of sublocale in the duality of T D -spaces. A sublocale S of a frame L is a D-sublocale if and only if the corresponding localic map preserves the property of being a covered prime. It is shown that for a frame L the system of those sublocales which are also D-sublocales form a dense sublocale S D (L) of the coframe S(L) of all its sublocales. It is also shown that the spatialization sp D [S D (L)] of S D (L) consists precisely of those D-sublocales of L which are T D -spatial. Additionally, frames such that we have S D (L) ∼ = P(pt D (L)) -that is, those such that D-sublocales perfectly represent subspaces -are characterized as those T D -spatial frames such that S D (L) is the Booleanization of S(L).

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Separation axioms and frame representation of some topological facts

Cited by 23 publications

References 8 publications

Monotone normality and stratifiability from a pointfree point of view

Monotone normality and stratifiability from a pointfree point of view

Revisiting the relation between subspaces and sublocales

The coframe of D-sublocales of a locale and the $T_D$ duality

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