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On open well-filtered spaces [article]

Chong Shen, Xiaoyong Xi, Xiaoquan Xu, Dongsheng Zhao
2020 arXiv   pre-print
The main results we proved include (1) every well-filtered space is an open well-filtered space; (2) every core-compact open well-filtered space is sober.  ...  We introduce and study a new class of T_0 spaces, called open well-filtered spaces.  ...  We prove that (i) every well-filtered space is an open well-filtered space, and (ii) every core-compact open well-filtered space is sober.  ... 
arXiv:2001.07397v4 fatcat:vfilnixcejhttjmsar76ptqese

On open well-filtered spaces

Chong Shen, Xiaoyong Xi, Xiaoquan Xu, Dongsheng Zhao
2020 Logical Methods in Computer Science  
The main results we proved include (1) every well-filtered space is an open well-filtered space; (2) every core-compact open well-filtered space is sober.  ...  We introduce and study a new class of $T_0$ spaces, called open well-filtered spaces.  ...  We prove that (i) every well-filtered space is an open well-filtered space, and (ii) every core-compact open well-filtered space is sober.  ... 
doi:10.23638/lmcs-16(4:18)2020 fatcat:dfedoqytabemfpwspgpedbxwqy

Some open problems on well-filtered spaces and sober spaces [article]

Xiaoquan Xu, Dongsheng Zhao
2020 arXiv   pre-print
In this paper, we shall present a brief summarising survey on some of such development. Furthermore, we shall pose and illustrate some open problems on well-filtered spaces and sober spaces.  ...  In the past few years, the research on sober spaces and well-filtered spaces has got some breakthrough progress.  ...  Is a retract of a co-sober space co-sober? Question 9.7. Is the product space of a family of co-sober spaces co-sober? Question 9.8. Is Co-Sob reflective in Top 0 ? References  ... 
arXiv:2005.00539v1 fatcat:4g76l7pyg5hl7cqioii3omx5vm

On Function Spaces Related to H-sober Spaces [article]

Meng Bao, Xiaoyuan Zhang, Xiaoquan Xu
2022 arXiv   pre-print
One immediate corollary is that for a T_0 space X, Y is a sober space (resp., d-space, well-filtered space) iff the function space ℂ(X, Y) equipped with the topology of pointwise convergence is a sober  ...  space (resp., d-space, well-filtered space) iff the function space ℂ(X, Y) equipped with the the Isbell topology is a sober space (resp., d-space, well-filtered space).  ...  Indeed, suppose that Y is a well-filtered space for which the Scott space ΣY is not well-filtered and X is the topological space with only one point.  ... 
arXiv:2204.08703v1 fatcat:i64ncgn57rf2zbf72uh4n4awei

ω-Rudin spaces, well-filtered determined spaces and first-countable spaces [article]

Xiaoquan Xu, Chong Shen, Xiaoyong Xi, Dongsheng Zhao
2020 arXiv   pre-print
closed subset in a first-countable ω-well-filtered space is countably-directed; (vi) every first-countable ω-well-filtered ω^∗-d-space is sober.  ...  We investigate some versions of d-space, well-filtered space and Rudin space concerning various countability properties.  ...  [3] ): sobriety ⇒ well-filteredness ⇒ d-space.  ... 
arXiv:2008.11023v1 fatcat:4od543uir5hcdmqsm23kqej3mq

Some remarks on Doitchinov completeness

Hans-Peter A. Künzi, Salvador Romaguera
1996 Topology and its Applications  
We observe that the well-monotone (open covering) quasiuniformity of each topological space is left K-complete.  ...  On the other hand, we exhibit an example of a topological space the fine quasiuniformity of which is not D-complete.  ...  Let or be a Cauchy filter with respect to W on X and let G be a maximal open filter on X containing all open members of .T.  ... 
doi:10.1016/s0166-8641(96)00044-2 fatcat:fwngienyirgz7otv4lmuxsmqgy

Page 6544 of Mathematical Reviews Vol. , Issue 94k [page]

1994 Mathematical Reviews  
Summary: “Balanced collections of open ultrafilters, closed ultrafil- ters, minimal prime open filters, and minimal prime closed filters are used to define balanced open filters, balanced closed filters  ...  , minimal balanced open filters, and minimal balanced closed fil- ters, respectively.  ... 

Subcompactness and the baire category theorem

J. de Groot
1963 Indagationes Mathematicae (Proceedings)  
all open sets is used.  ...  Roughly speaking, subcompactness of a space is a weak form of compactness relative to some open base of the space; subcompactness in completely regular spaces is identical to compactness if the base of  ...  Indeed, the subcompactness follows directly from one of the well known compactness criteria.  ... 
doi:10.1016/s1385-7258(63)50076-6 fatcat:h7xgmlghffhtvgpf4qdlj77qhm

Compact semilattices with open principal filters

Oleg V. Gutik, M. Rajagopalan, K. Sundaresan
2002 Journal of the Australian Mathematical Society  
Structure of topological semilattices on the one-point Alexandroff compactification of an uncountable discrete space and linearly ordered compact semilattices with open principal filters are researched  ...  A locally compact semilattice with open principal filters is a zero-dimensional scattered space.  ...  So there Some classes of compact semilattices with open principal filters A semilattice E is called linearly ordered (well-ordered) if the multiplication induces on E a linear order (a well-order).  ... 
doi:10.1017/s1446788700036776 fatcat:6bh72r2frrbupdkkmrvxf2rdhu

Hofmann-Mislove through the Lenses of Priestley [article]

G. Bezhanishvili, S. Melzer
2022 arXiv   pre-print
The next definition is well known (see, e.g., [11] ). Definition 3.1. Let L be a frame. A filter F of L is Scott-open if S ∈ F implies T ∈ F for some finite T ⊆ S. Let X be a sober space.  ...  Thus, the topology on Y is the restriction of the open upset topology on X (see [1, Lem. 5.3]).  ...  Proof. (1) It is well known and easy to see that a Scott-open filter is completely prime iff it is prime.  ... 
arXiv:2205.10891v1 fatcat:m6lbxqsh6nhrlo7qwg7aem7pxe

On "Subcompactness and the Baire category theorem"

Isidore Fleischer
1979 Indagationes Mathematicae (Proceedings)  
Of course the union of an increasing sequence of regular filter bases is again one.  ...  A regular Hausdorff space is subcompact relative to an open base {U} if every regular filter base drawn from {U} has non-void intersection: a regular filter base is a non-empty collection of non-void sets  ... 
doi:10.1016/s1385-7258(79)80003-7 fatcat:76jlodtws5fidcytprszy5r77q

Spaces with unique Hausdorff extensions

Douglas D. Mooney
1995 Topology and its Applications  
This space is also an example of a space with exactly one free prime open filter which gives an answer to a question asked by J. Pelant, P. Simon, and J. Vaughan.  ...  If Sr is a free open filter on X, then P(g) is a compact subset of aX\X. Conversely, if A is a compact subset of aX\X, then f-j A is a free open filter and P( n A) = A.  ...  Each open filter on a space is the intersection of the prime open filters on that space that contain it.  ... 
doi:10.1016/0166-8641(94)00031-w fatcat:zt73d7swdzandews5cmlv2zufu

Adjoint product and hom functors in general topology

Peter Wilker
1970 Pacific Journal of Mathematics  
The ordinary product topology on R X S and the compact-open topology on T s will be given their proper setting in this context.  ...  The well known natural equivalence [RxS,T] = [R, T s ], valid in the category of sets and set mappings, can be derived in various ways in the category of topological spaces and continuous maps, provided  ...  Given S, the letter @ will be used to describe the space of open sets of S as well as the corresponding lattice, g, ©, will be used for filters on @, ^ gf, ^C, 3ίΓ, for families of such filters.  ... 
doi:10.2140/pjm.1970.34.269 fatcat:5ktizr6lenhofiiocdlwrxiow4

Bicompleteness of the fine quasi-uniformity

Hans-Peter A. Künzi, Nathalie Ferrario
1991 Mathematical proceedings of the Cambridge Philosophical Society (Print)  
Let X be a topological space and let ~V be a compatible quasi-uniformity on X that is finer than the Pervin quasi-uniformity 0> of X. (a) Let ?F be a ~V*-Cauchy filter on X, and let x eX. Then !  ...  We obtain several conditions under which the semi-continuous quasi-uniformity of a topological space is bicomplete and observe that the well-monotone covering quasiuniformity of a topological space is  ...  of subsets of a topological space X is called well-monotone (see [16] , p. 20) provided that the partial order £ of set inclusion is a well-order on jSf.  ... 
doi:10.1017/s0305004100069644 fatcat:jr7npqpftzfolctv34ti5fas54

First countability, ω-well-filtered spaces and reflections [article]

Xiaoquan Xu, Chong Shen, Xiaoyong Xi, Dongsheng Zhaod
2019 arXiv   pre-print
One immediate corollary is that every core compact well-filtered space is sober, answering Jia-Jung problem with a new method.  ...  Therefore, a first countable T_0 space X is sober iff X is well-filtered iff X is an ω-well-filtered d-space.  ...  Proof. (1) ⇒ (2): Suppose that X is an ω-well-filtered space. For any countable K ∈ K(X), by the ω-well-filteredness of X, K ∈ K(X). Therefore, by Lemma 2.12, K(X) is an ω-dcpo.  ... 
arXiv:1911.13201v1 fatcat:y62hxiq4vfbo3a3leplm2clkxy
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