Kenneth Kunen
There is a locally compact Hausdorff space of weight aleph_omega which is linearly Lindelof and not Lindelof. This improves an earlier result, which produced such a space of weight beth_omega.
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We call a space $X$ {\it weakly linearly Lindelöf} if for any family $\mathcal{U}$ of non-empty open subsets of $X$ of regular uncountable cardinality $κ$, there exists a point $x\in X$ such that every neighborhood of $x$ meets $κ$-many elements of $\mathcal{U}$. We also introduce the concept of {\it almost discretely Lindelöf} spaces as the ones in which every discrete subspace can be covered by a Lindelöf subspace. We prove that, in addition to linearly Lindelöf spaces, both weakly Lindelöf spaces and almost discretely Lindelöf spaces are weakly linearly Lindelöf. The main result of the pape…
The main results of this note are: It is consistent that every subparacompact space $X$ of size $ω_1$ is a $D$-space; If there exists a Michael space, then all productively Lindelöf spaces have the Menger property, and, therefore, are $D$-spaces; and Every locally $D$-space which admits a $σ$-locally finite cover by Lindelöf spaces is a $D$-space.
We examine locally compact normal spaces in models of form PFA(S)[S], in particular characterizing paracompact, countably tight ones as those which include no perfect pre-image of omega_1 and in which all separable closed subspaces are Lindelof.
A space is od-compact (resp. od-Lindelöf) provided any cover by open dense sets has a finite (resp. countable) subcover. We first show with simple examples that these properties behave quite poorly under finite or countable unions. We then investigate the relations between Lindelöfness, od-Lindelöfness and linear Lindelöfness (and similar relations with `compact'). We prove in particular that if a $T_1$ space is od-compact, then the subset of its non-isolated points is compact. If a $T_1$ space is od-Lindelöf, we only get that the subset of its non-isolated points is linearly Lindelöf. Though,…
We give several new bounds for the cardinality of a Hausdorff topological space $X$ involving the weak Lindelöf degree $wL(X)$. In particular, we show that if $X$ is extremally disconnected, then $|X|\leq 2^{wL(X)πχ(X)ψ(X)}$, and if $X$ is additionally power homogeneous, then $|X|\leq 2^{wL(X)πχ(X)}$. We also prove that if $X$ is an almost Lindelöf space with a strong $G_δ$-diagonal of rank 2, then $|X|\leq 2^{\aleph_0}$; that if $X$ is a star-cdc space with a $G_δ$-diagonal of rank 3, then $|X| \le 2^{\aleph_0}$; and if $X$ is any normal star-cdc space $X$ with a $G_δ$-diagonal of rank 2, the…
We construct locally Lindelöf scattered P-spaces (LLSP spaces, in short) with prescribed widths and heights under different set-theoretic assumptions. We prove that there is an LLSP space of width $ω_1$ and height $ω_2$ and that it is relatively consistent with ZFC that there is an LLSP space of width $ω_1$ and height $ω_3$. Also, we prove a stepping up theorem that, for every cardinal $λ\geq ω_2$, permits us to construct from an LLSP space of width $ω_1$ and height $λ$ satisfying certain additional properties an LLSP space of width $ω_1$ and height $α$ for every ordinal $α< λ^+$. Then, we obt…
It is consistent with MA plus not CH that there is a locally connected hereditarily Lindelof compact space which is not metrizable.
Cardinal Sequences of Lindelöf scattered P-spaces
该研究继续探究LLSP空间的基数序列，提出了从锥系统和特定偏序构造LLSP空间的方法，确定了其基数序列的限制，并给出了序列$\langle κ_α: α< ω_2 \rangle$作为LLSP空间基数序列的必要条件与充分条件。
A topological space $X$ is strongly $D$ if for any neighbourhood assignment $\{U_x:x\in X\}$, there is a $D\subseteq X$ such that $\{U_x:x\in D\}$ covers $X$ and $D$ is locally finite in the topology generated by $\{U_x:x\in X\}$. We prove that $\diamondsuit$ implies that there is an $HFC_w$ space in $2^{ω_1}$ (hence 0-dimensional, Hausdorff and hereditarily Lindelöf) which is not strongly $D$. We also show that any $HFC$ space $X$ is dually discrete and if additionally, countable sets have Menger closure then $X$ is a $D$-space.
Tkachuk and Wilson proved that a regular first countable cellular-compact space has cardinality not exceeding the continuum. In the same paper they asked if this result continues to hold for Hausdorff spaces. Xuan and Song considered the same notion and asked if every cellular-compact space is weakly Lindelof. We answer the last question for first countable spaces. As a byproduct of this result, we present a somewhat different proof of Tkachuk and Wilson theorem, valid for the wider class of Urysohn spaces. The result actually holds for a class of spaces in between cellular-compact and cellula…
A topological space $(X,τ)$ is called a locally LC-space if every point of $X$ has a neighborhood $U$ such that every Lindelöf subset of $(U,τ|U)$ is a closed subset of $(U,τ|U)$. The aim of this paper is to continue the study of locally LC-spaces.
We obtain several results and examples concerning the general question ``When must a space with a small diagonal have a G_delta-diagonal?". In particular, we show (1) every compact metrizably fibered space with a small diagonal is metrizable; (2) there are consistent examples of regular Lindelof (even hereditarily Lindelof) spaces with a small diagonal but no G_delta-diagonal; (3) every first-countable hereditarily Lindelof space with a small diagonal has a G_delta-diagonal; (4) assuming CH, every Lindelof Sigma-space with a small diagonal has a countable network; (5) whether countably compact…
We characterize, in terms of $X$, extensional dimension of the Stone-Čech corona $βX \setminus X$ of locally compact and Lindelöf space $X$. The non-Lindelöf case case is also settled in terms of extending proper maps with values in $I^τ\setminus L$, where $L$ is a finite complex. Further, for a finite complex $L$, an uncountable cardinal $τ$ and a $Z_τ$-set $X$ in the Tychonov cube $I^τ$ we find necessary and sufficient condition, in terms of $I^τ\setminus X$, for $X$ to be in the class $\operatorname{AE}([L])$. We also introduce a concept of a proper absolute extensor and characterize the pr…
We give a general closing-off argument in Theorem 2.1 from which several corollaries follow, including (1) if $X$ is a locally compact Hausdorff space then $|X|\leq 2^{wL(X)ψ(X)}$, and (2) if $X$ is a locally compact power homogeneous Hausdorff space then $|X|\leq 2^{wL(X)t(X)}$. The first extends the well-known cardinality bound $2^{ψ(X)}$ for a compactum $X$ in a new direction. As $|X|\leq 2^{wL(X)χ(X)}$ for a normal space $X$ [3], this enlarges the class of known Tychonoff spaces for which this bound holds. In 2.10 we give a short, direct proof of (1) that does not use 2.1. Yet 2.1 is broad…
We show that for a normal locally-${\mathscr P}$ space $X$ (where ${\mathscr P}$ is a topological property subject to some mild requirements) the subset $C_{\mathscr P}(X)$ of $C_b(X)$ consisting of those elements whose support has a neighborhood with ${\mathscr P}$, is a subalgebra of $C_b(X)$ isometrically isomorphic to $C_c(Y)$ for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$ is explicitly constructed as a subspace of the Stone--Čech compactification $βX$ of $X$ and contains $X$ as a dense subspace. Under certain conditions, $C_{\mathscr P}(X)$ coinci…
A space $X$ is Eberlein-Grothendieck if $X\subset C_p(K)$ for some compact space $K.$ In this paper we address the problem of whether such a space $X$ is $σ$-discrete whenever it is scattered. We show that if $w(K)\leqω_1$ then $X$ is $σ$-dicrete whenever $X$ has height $ω_1$ and it is locally compact or locally countable. It is also proved that every Lindelöf Čech-complete scattered space is $σ$-compact.
We introduce notions of nearly good relations and N-sticky modulo a relation as tools for proving that spaces are D-spaces. As a corollary to general results about such relations, we show that C_p(X) is hereditarily a D-space whenever X is a Lindelöf Σ-space. This answers a question of Matveev, and improves a result of Buzyakova, who proved the same result for X compact. We also prove that if a space X is the union of finitely many D-spaces, and has countable extent, then X is linearly Lindelöf. It follows that if X is in addition countably compact, then X must be compact. We also show that Co…
New results about Q and $Δ$-spaces
证明存在可测基数与存在拥挤的Baire T1 Δ-空间等几个命题是等一致的，并给出关于Lindelöf Q-空间的新结果。
Gelfand-Naimark-Stone duality provides an algebraic counterpart of compact Hausdorff spaces in the form of uniformly complete bounded archimedean $\ell$-algebras. In [4] we extended this duality to completely regular spaces. In this article we use this extension to characterize normal, Lindëlof, and locally compact Hausdorff spaces. Our approach gives a different perspective on the classical theorems of Katětov-Tong and Stone-Weierstrass.
We apply the theory of infinite two-person games to two well-known problems in topology: Suslin's Problem and Arhangel'skii's problem on $G_δ$ covers of compact spaces. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable and 2) in every compact space satisfying the game-theoretic version of the weak Lindelöf property, every cover by $G_δ$ sets has a continuum-sized subcollection whose union is $G_δ$-dense.
We introduce the algebraic entropy for continuous endomorphisms of locally linearly compact vector spaces over a discrete field, as the natural extension of the algebraic entropy for endomorphisms of discrete vector spaces. We show that the main properties of entropy continue to hold in the general context of locally linearly compact vector spaces, in particular we extend the Addition Theorem.
Eventually Constant and stagnating functions in non-Lindelöf spaces
研究非林德洛夫空间中函数性质，引入推广林德洛夫性的四个性质$\mathsf{P}(X,Y)$，研究其与其他拓扑性质关系，给出如特定树子空间、流形等相关结果示例，部分结果是对前人工作的重新表述。
In analogy to the topological entropy for continuous endomorphisms of totally disconnected locally compact groups, we introduce a notion of topological entropy for continuous endomorphisms of locally linearly compact vector spaces. We study the fundamental properties of this entropy and we prove the Addition Theorem, showing that the topological entropy is additive with respect to short exact sequences. By means of Lefschetz Duality, we connect the topological entropy to the algebraic entropy in a Bridge Theorem.
A space is called linearly H-closed iff any chain cover possesses a dense member. This property lies strictly between feeble compactness and H-closedness. While regular H-closed spaces are compact, there are linearly H-closed spaces which are even collectionwise normal and Fréchet-Urysohn. We give examples in other classes, and ask whether there is a first countable normal linearly H-closed non-compact space in ZFC. We show that PFA implies a negative answer if the space is moreover either locally separable or locally compact and locally ccc. Ostaszewski space (built with $\diamondsuit$) is an…
Contents: 1. Editor's note 2. Research announcements 2.1. Pseudocompact group topologies with no infinite compact subsets 2.2. Selective coideals on (FIN[1]k) 2.3. Entire functions mapping uncountable dense sets of reals onto each other monotonically 2.4. Effective refining of Borel coverings 2.5. Symmetry and colorings: Some results and open problems 2.6. Many partition relations below density 2.7. Lindelof indestructibility, topological games and selection principles 2.8. Locally precompact groups: (Local) realcompactness and connectedness 2.9. The group Aut is Roelcke precompact 2.10. An in…
A space $Y$ is called an {\em extension} of a space $X$ if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {\em equivalent} if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Y\leq Y'$ if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ of $X$ is called a {\em one-point extension} if $Y\backslash X$ is a singleton. An extension $Y$ of $X$ is called {\em first-countable} if $Y$ is first-countable at points of $Y\backslash X$. Let ${\mathca…
Weight of convex compact spaces and their boundaries
研究凸紧空间及其边界的权重关系，证明边界具有可数网络时空间可度量，且 Lindelöf $Σ$-边界下网络权重与权重相等。
An open chain cover $\{U_α: α\inκ\}$ ($κ$ a cardinal) of a space $X$ is a systematic cover if the closure of $U_α$ is contained in $U_β$ when $α<β$, and $X$ is Type I if $κ=ω_1$ and the closure of each $U_α$ is Lindelöf. A closed subspace $D\subset X$ is narrow in $X$ if for each systematic cover $\{V_α: α\inω_1\}$ of $X$, either there is $α$ such that $D$ is included in $V_α$, or the closure of $V_α$ intersected with $D$ is Lindelöf for each $α$. Taking systematic covers given by preimages by $s$ of $[0,α)$ for a continuous $s: X\to \mathbb{L}_{\ge 0}$ (where $ \mathbb{L}_{\ge 0}$ is the long…
We prove that every infinite-dimensional (locally convex) linear topological space that can be expressed as a direct limit of finite-dimensional metrizable compacta is (linearly) homeomorphic to the space $R^\infty=\dlim R^n$.
Under $(\diamondsuit^*)$ we construct a locally countable, locally compact, 0-dimensional $T_2$ space $X$ of size $\omg$ which is aD however not even linearly D. This consistently answers a question of Arhangel'skii, whether aD implies D. Furthermore we answer two problems concerning characterization of linearly D-spaces, raised by Guo and Junnila.
A coarse space $X$, endowed with a linear order compatible with the coarse structure of $X$, is called linearly ordered. We prove that every linearly ordered coarse space $X$ is locally convex and the asymptotic dimension of $X$ is either $0$ or $1$. If $X$ is metrizable then the family of all right bounded subsets of $X$ has a selector.
Given a class $\mathcal P$ of Banach spaces, a locally convex space (LCS) $E$ is called {\em multi-$\mathcal P$} if $E$ can be isomorphically embedded into a product of spaces that belong to $\mathcal P$. We investigate the question whether the free locally convex space $L(X)$ is strongly nuclear, nuclear, Schwartz, multi-Hilbert or multi-reflexive. If $X$ is a Tychonoff space containing an infinite compact subset then, as it follows from the results of \cite{Aus}, $L(X)$ is not nuclear. We prove that for such $X$ the free LCS $L(X)$ has the stronger property of not being multi-Hilbert. We ded…
If a Tychonoff space $X$ is dense in a Tychonoff space $Y$, then $Y$ is called a Tychonoff extension of $X$. Two Tychonoff extensions $Y_1$ and $Y_2$ of $X$ are said to be equivalent, if there exists a homeomorphism $f:Y_1\rightarrow Y_2$ which keeps $X$ pointwise fixed. This defines an equivalence relation on the class of all Tychonoff extensions of $X$. We identify those extensions of $X$ which belong to the same equivalence classes. For two Tychonoff extensions $Y_1$ and $Y_2$ of $X$, we write $Y_2\leq Y_1$, if there exists a continuous function $f:Y_1\rightarrow Y_2$ which keeps $X$ pointw…
We characterize Ascoli spaces by showing that a Tychonoff space $X$ is Ascoli iff the canonical map from the free locally convex space $L(X)$ over $X$ into $C_k\big(C_k(X)\big)$ is an embedding of locally convex spaces. We prove that an uncountable direct sum of non-trivial locally convex spaces is not Ascoli. If a $c_0$-barrelled space $X$ is weakly Ascoli, then $X$ is linearly isomorphic to a dense subspace of $\mathbb{R}^Γ$ for some $Γ$. Consequently, a Fréchet space $E$ is weakly Ascoli iff $E=\mathbb{R}^N$ for some $N\leqω$. If $X$ is a $μ$-space and a $k$-space (for example, metrizable),…
Quotient homomorphisms of Topological MV-Algebras and Applications
本文证明拓扑MV-代数中理想I的商映射是连续开映射，且若I紧则商映射完美，并应用于研究紧性、局部紧性等三空间性质。
The Bishop property ($\symbishop$), introduced recently by K.P. Hart, T. Kochanek and the first-named author, was motivated by Pełczy{ń}ski's classical work on weakly compact operators on $C(K)$-spaces. This property asserts that certain chains of functions in said spaces, with respect to a particular partial ordering, must be countable. There are two versions of ($\symbishop$): one applies to linear operators on $C(K)$-spaces and the other to the compact Hausdorff spaces themselves. We answer two questions that arose after ($\symbishop$) was first introduced. We show that if $\mathscr{D}$ is …
The paper deals with Ascoli spaces $C_p(X)$ and $C_k(X)$ over Tychonoff spaces $X$. The class of Ascoli spaces $X$, i.e. spaces $X$ for which any compact subset $K$ of $C_k(X)$ is evenly continuous, essentially includes the class of $k_{\mathbb R}$-spaces. First we prove that if $C_p(X)$ is Ascoli, then it is $κ$-Fréchet-Urysohn. If $X$ is cosmic, then $C_p(X)$ is Ascoli iff it is $κ$-Fr'echet-Urysohn. This leads to the following extension of a result of Morishita: If for a Čech-complete space $X$ the space $C_p(X)$ is Ascoli, then $X$ is scattered. If $X$ is scattered and stratifiable, then $…
The paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $Δ$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_0$ which is isomorphic to a closed subspace of a space with a basis and $C(I, X)$ is linearly homeomorphic to $C(Δ, X)$, then $X$ is locally convex, i.e., a Banach space. It is shown that Kalton result is sharp by exhibiting non locally convex quasi Banach spaces X with a basis for which $C(I, X)$ and $C(Δ, X)$ are isomorphic. Our examples are rather specific and actually in all cases X is isomorphic to $C(Δ, X)$ …
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