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On U_c-admitting spaces and U_c-completions of T_0 spaces
Kangming Lu, Qingguo Li, Shengwen Wang, Rongqi Xiao
TL;DR
The relationship between U_s-admitting and well-filtered spaces leaves open whether their gap can be refined. The paper introduces U_c-admitting spaces and, through the H-sober framework, establishes their place within H-sobriety and key structural results.
Problem
The relationship between U_s-admitting and well-filtered spaces leaves open whether their gap can be refined.
Source: §1 · p. 1Method
The paper introduces U_c-admitting spaces, adopts the H-sober framework, and uses U_c-admitting determined sets to construct a completion.
Source: §1 · pp. 1–2Results
Every U_c-admitting space is H-sober; U_c-Sob is complete and reflective in Top_0, and X is U_c-admitting exactly when its Smyth power space is.
Source: §1 · p. 2Takeaways & Limitations
The standard U_c-completion provides a universal embedding of any T_0 space into a U_c-admitting space.
Source: §1 · p. 2For a T_0 space, U_c-admittingness together with local compactness or core compactness does not imply sobriety.
Source: §3 · p. 11
Abstract
from arXiv · showhide
In this paper, we introduce a new class of spaces, termed U_c-admitting spaces, and investigate their relationships with U_s-admitting spaces and well-filtered spaces. Adopting the perspective of H-sobriety, we show that every U_c-admitting space is H-sober, so that U_c-admitting spaces arise as a special case of H-sober spaces. Consequently, the category U_c-Sob of U_c-admitting spaces and continuous maps is complete and forms a reflective subcategory of Top_0. In addition, we prove that a T_0 space X is U_c-admitting if and only if its Smyth power space is U_c-admitting. Finally, we present the standard U_c-completion of an arbitrary T_0 space.
1. Introduction
The paper asks whether the gap between U_s-admitting and well-filtered spaces can be refined, introducing U_c-admitting spaces through a weakened compactness condition. It develops their structural and categorical properties and constructs a U_c-completion for T_0 spaces.
1. Introduction
- The paper introduces U_c-admitting spaces to refine the open gap between U_s-admitting and well-filtered spaces.The earlier hierarchy places U_s-admitting spaces strictly between d-spaces and well-filtered spaces.Source: §1 · p. 1
- U_c-admittingness weakens well-filteredness by requiring weakly strongly compact saturated sets, rather than compact saturated sets, inside open neighbourhoods.The paper presents this change as the motivation for a new class of spaces.Source: §1 · p. 2
- In countable T_0 spaces, U_c-admittingness coincides with well-filteredness; combined with local countable compactness or first-countability, it implies sobriety.Source: §1 · p. 2
- U_c-sets form a class of H-sets, and U_c-admitting spaces form a complete category reflective in Top_0; U_c-admittingness is also preserved and reflected by the Smyth power space.Source: §1 · p. 2
- The paper constructs a standard U_c-completion that universally embeds any T_0 space into a U_c-admitting space.The construction uses U_c-admitting determined sets.Source: §1 · p. 2
2. Preliminaries
This section establishes terminology and background for comparing d-spaces, well-filtered spaces, sober spaces, and U_s-admitting spaces. It also gives examples distinguishing U_s-admittingness from related properties.
2. Preliminaries
- A T_0 space is sober when every irreducible closed subset is the closure of a unique point.Irreducibility means a nonempty set contained in a union of two closed sets must be contained in one of them.Source: §2 · p. 3
- Well-filteredness requires some member of every filtered family of compact saturated sets to lie inside an open set containing their intersection.The section also relates well-filteredness to KF-sets: it is equivalent to every KF-set having the closure of a unique point.Source: §2 · p. 3
- The Hoare power space uses the lower Vietoris topology, while the Smyth power space uses the upper Vietoris topology on compact saturated sets.Source: §2 · p. 4
- U_s-admitting spaces require every U_s-set to be a singleton, where U_s-sets arise as minimal closed sets avoiding a filtered family of upward closures of strongly compact sets.In T_1 spaces, strongly compact sets are exactly finite sets.Source: §2 · p. 5
- The examples show that U_s-sets need not be directed or KF-sets, and U_s-admittingness does not imply well-filteredness.The cofinite topology on the natural numbers is U_s-admitting but not well-filtered; another example supplies a U_s-set that is not directed.Source: §2 · pp. 5–6
- The cofinite natural-number space is locally compact, core compact, and first-countable, yet it is not sober.This example shows why those properties alone do not establish sobriety for U_s-admitting spaces.Source: §2 · p. 6
3. Uc-admitting space
This section develops the notion of U_c-admitting spaces by defining weak strong compactness and U_c-sets, examining their properties and relations to other space classes, and giving counterexamples that separate those classes. It also establishes sobriety for A1 U_c-admitting spaces, where every irreducible closed set is the closure of a singleton.
3. Uc-admitting space
- Weakly strongly compact sets are preserved by saturation, finite unions, continuous images, and closed intersections; in T1 spaces they are exactly countable compact sets.The image of a Uc-set under a continuous map is also a Uc-set.Source: §3 · p. 7
- A Uc-admitting T0 space requires every Uc-set, defined via a filtered family of weakly strongly compact saturated sets, to be a singleton.Weak strong compactness means compactness plus the existence, inside every open neighborhood, of a countable set whose upward closure contains the compact set.Source: §3 · pp. 7–8
- Sober spaces imply well-filtered spaces, which imply Uc-admitting spaces, which imply Us-admitting spaces and d-spaces.The implications are strict at the Uc-admitting boundary: the T1 space Ncof is Us-admitting but not Uc-admitting.Source: §3 · p. 8
- Uc-admission is equivalent to filtered-family, closed-set, and pointwise open-set tests involving Qc(X) and Uc-sets.These formulations include the requirement that if the intersection of a filtered family in Qc(X) lies in an open set, then some member of the family lies in that open set.Source: §3 · p. 9
- For countable T0 spaces, being Uc-admitting is equivalent to being well-filtered.The proof uses that every compact saturated set in a countable space is weakly strongly compact.Source: §3 · p. 10
- The space (X, τ) is Uc-admitting but not well-filtered: the filtered family {[a, ω1) : a ∈ X} consists of compact sets in (X, τ) and has empty intersection.Source: §3 · p. 11
- Every locally countably compact Uc-admitting space is sober.The section also notes that Uc-admission together with local compactness, or with core compactness, does not by itself imply sobriety.Source: §3 · pp. 11–12
- Every A1 U_c-admitting space is sober, with each irreducible closed set equal to the closure of a singleton.The proof establishes that irreducible closed sets are directed; because U_c-admitting spaces are d-spaces, they are closures of singletons.Source: §3 · p. 13
- First countability supplies decreasing neighborhood bases for constructing sequences that establish directedness of irreducible closed sets.The proof forms a filtered family of locally weakly strongly compact sets and applies the U_c-admitting property to obtain the needed intersection.Source: §3 · p. 13
4. H-sober approach to Uc-admitting spaces
The H-sober framework identifies U_c-admitting spaces as U_c-sober spaces and establishes their categorical properties. It also yields closure and product results, together with an equivalence between a T_0 space and its U_c-Smyth power space.
H-sobriety
- U_c-admitting spaces are exactly U_c-sober spaces, and U_c-sets form an R-subset system.This places the class within the H-sober framework used to derive its categorical properties.Source: §4 · p. 14
Stability properties
- Nonempty closed subspaces and nonempty saturated subspaces of a U_c-admitting space, as well as its retracts, are U_c-admitting.Source: §4 · p. 15
Products
- A product is U_c-admitting if and only if each of its factors is U_c-admitting.Source: §4 · p. 15
Categorical properties
- The category U_c-Sob is complete and reflective in Top_0.These categorical properties follow from the R-subset-system framework.Source: §4 · p. 15
Smyth power spaces
- For a T_0 space X, its U_c-Smyth power space is U_c-admitting if and only if X is U_c-admitting.The theorem also characterizes both conditions as equivalent to the power space being a d-space.Source: §4 · p. 15
5. The Uc-completion of T0 space
The paper defines a universal U_c-completion for T_0 spaces, extending the completion constructions described for other classes of spaces. Its standard construction uses U_c-admitting determined sets and yields a U_c-admitting space.
5. The Uc-completion of T0 space
- A UCD-set is a subset whose image under every continuous map into a U_c-admitting space is represented by a unique point.The family of closed UCD-sets of X forms the underlying family used in the standard construction.Source: §5 · p. 17
- The U_c-completion extends the universal-property pattern used for D-completions, well-filterifications, soberifications, and U_s-completions.For every continuous map from X to a U_c-admitting space Z, there is a unique continuous map from the completion Y that factors it through η.Source: §5 · pp. 16–17
- The standard completion equips the family of closed UCD-sets with a topology and a map η, yielding the U_c-completion of X.The supplied construction passages introduce the family and topology but do not specify their full formulas.Source: §5 · p. 18
- The map η sends each x in X to ↓x, and the proof verifies that η is continuous and that maps out of X extend uniquely.The proof establishes uniqueness by showing the extension and any competing map agree on the dense image η(X).Source: §5 · p. 18
- The constructed space Y is U_c-admitting because every closed U_c-set in Y is the closure of a singleton.The proof represents such a set as □C for a closed UCD-set C, then uses that C belongs to Y.Source: §5 · pp. 18–19