MinimalT0-spaces and minimalTD-spaces

Larson, Robert E.

doi:10.2140/pjm.1969.31.451

Cited by 11 publications

(4 citation statements)

References 5 publications

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“…The topological analogue of an extension of an ordering is a coarser T 0 topology, and therefore the topological analogue of a linear extension (namely, a maximal ordering extending the original ordering) should be a minimal T 0 topology coarser than the original T 0 topology. However, Larson [24,Example 6] showed that such minimal T 0 topologies may fail to exist. This is true even in the realm of Noetherian spaces: Larson's example is R with its cofinite topology, and every set is Noetherian in its cofinite topology.…”

Section: Staturementioning

confidence: 99%

Statures and Sobrification Ranks of Noetherian Spaces

Jean¹,

Laboureix²

2021

Preprint

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There is a rich theory of maximal order types of well-partial-orders (wpos), pioneered by de Jongh and Parikh (1977) and Schmidt (1981). Every wpo is Noetherian in its Alexandroff topology, and there are more; this prompts us to investigate an analogue of that theory in the wider context of Noetherian spaces.The notion of maximal order type does not seem to have a direct analogue in Noetherian spaces per se, but the equivalent notion of stature, investigated by Blass and Gurevich (2008) does: we define the stature ||X|| of a Noetherian space X as the ordinal rank of its poset of proper closed subsets. We obtain formulas for statures of sums, of products, of the space of words on a space X, of the space of finite multisets on X, in particular. They confirm previously known formulas on wpos, and extend them to Noetherian spaces.The proofs are, by necessity, rather different from their wpo counterparts, and rely on explicit characterizations of the sobrifications of the corresponding spaces, as obtained by Finkel and the first author (2020).We also give formulas for the statures of some natural Noetherian spaces that do not arise from wpos: spaces with the cofinite topology, Hoare powerspaces, powersets, and spaces of words on X with the socalled prefix topology.Finally, because our proofs require it, and also because of its independent interest, we give formulas for the ordinal ranks of the sobrifications of each of those spaces, which we call their dimension.

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Section: Staturementioning

confidence: 99%

Statures and Sobrification Ranks of Noetherian Spaces

Jean¹,

Laboureix²

2021

Preprint

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“…and (s, a) [9] and Pahk [15] that a T 0 -topological space (X, ST) is minimal T Q iff {-{£}: x E X} U {X} is a base for 3 and finite unions of point closures are point closures.…”

Section: Fi(r)= V(r) Similarly Minimal T P Is Order-induced Iff Fi(mentioning

confidence: 99%

“…Larson [9] has also proved that a T D -topological space is minimal T D iff the topology is nested. Using the fact that when R& is linear the kernels are complements of derived sets, it is not difficult to prove as a corollary that a topological space (X, 3) is minimal T D iff R*r is linear and 3' is the kernel topology of R?.…”

Section: Fi(r)= V(r) Similarly Minimal T P Is Order-induced Iff Fi(mentioning

confidence: 99%

Order-induced topological properties

Andima¹,

Thron²

1978

Pacific J. Math.

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Each topology J on a set X may be associated with a preorder relation R& on X defined by (a, b) G R& iff every open set containing b contains a. Although the correspondence is many-to-one, there is always a least topology, ii(R), and a greatest topology, v(R), having a given preorder R. This leads to a natural correspondence between order properties and some topological properties and to the concept of an order-induced topological property. We show that a number of familiar topological properties (mostly lower separation axioms) are order-induced and also consider some new properties suggested by order properties. Let T p be an order-induced topological property with associated order property K p . We characterize minimal and maximal T p as follows: A topological space (X, 5~) is maximal T p iff 3~ = v{R^) and R* is minimal K p . With the imposition of a further condition on the class K p (satisfied by most properties under discussion), (X, 3~) is minimal T p iff SF = /X(JR^) and R°r is maximal K p . We apply these general theorems to a number of order-induced properties and conclude with an example to show that, for two particular properties, 3m ay be minimal T p even though R& is not maximal K p . Introduction.Correspondences between topologies and preorders on X similar to that assigning R^ to

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“…Given a topological space (X, T ), (X, T ) is -minimal Hausdorff if and only if it is Hausdorff and every open filterbase which has a unique adherent point is convergent to this point (see [5], [9], [10], [27], [31], [32], and [36]) -minimal T 1 if and only if T is the cofinite topology C on X -minimal regular if and only if it is regular and every regular filter-base which has a unique adherent point is convergent ([4], [8] [12], [19], [22], [26]) -minimal T D if and only if it is T D and nested ( [1], [12], [19], [22], [26] of X and some partition P of X such that P is simply associated with K and is associated with X \ K. ( [16]) -minimal T DD if and only if T = W K (P) ∨ (C ∩ I(K)) for some subset K of X and partition P of X such that P is simply associated with K and associated with X \ K ( [16]…”

Section: Introductionmentioning

confidence: 99%

Minimal convergence spaces

Kent¹,

Richardson²

1971

Trans. Amer. Math. Soc.

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Abstract.A topological space is TUD if the derived set of each point is the union of disjoint closed sets. We show that there is a minimal TUD space which is not just the Alexandroff topology on a linear order. Indeed the structure of the underlying partial order of a minimal TUD space can be quite complex. This contrasts sharply with the known results on minimality for weak separation axioms. 2000

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MinimalT₀-spaces and minimalT_D-spaces

Cited by 11 publications

References 5 publications

Statures and Sobrification Ranks of Noetherian Spaces

Statures and Sobrification Ranks of Noetherian Spaces

Order-induced topological properties

Minimal convergence spaces

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