Michael Levin
We prove that for every finite dimensional compact metric space $X$ there is an open continuous linear surjection from $C_p[0,1]$ onto $C_p(X)$. The proof makes use of embeddings introduced by Kolmogorov and Sternfeld in connection with Hilbert's 13th problem.
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A topological space $X$ is $κ$-Fréchet--Urysohn if for every open subset $U$ of $X$ and every $x\in \overline{U}$ there exists a sequence in $ U$ converging to $x$. We prove that every $κ$-Fréchet--Urysohn Tychonoff space $X$ is Ascoli. We apply this statement and some of known results to characterize the $κ$-Fréchet--Urysohn property in various important classes of locally convex spaces. In particular, answering a question posed in [7] we obtain that $C_p(X)$ is Ascoli iff $X$ has the property $(κ)$.
Let X be a non-degenerate, connected, locally path-connected metrizable space and Fin(X) be the hyperspace consisting of non-empty finite subsets in X endowed with the Vietoris topology. In this paper, we show that every compact set in Fin(X) is a strong Z-set.
We show that the consistency strength of $κ$ being $2^κ$-square compact is at least weak compact and strictly less than indescribable. This is the first known improvement to the upper bound of strong compactness obtained in 1973 by Hajnal and Juh{\' a}sz.
We extend to metrizable locally compact groups Rosenthal's theorem describing those Banach spaces containing no copy of $l_1$. For that purpose, we transfer to general locally compact groups the notion of interpolation ($I_0$) set, which was defined by Hartman and Ryll-Nardzewsky [25] for locally compact abelian groups. Thus we prove that for every sequence $\lbrace g_n \rbrace_{n<ω}$ in a locally compact group $G$, then either $\lbrace g_n \rbrace_{n<ω}$ has a weak Cauchy subsequence or contains a subsequence that is an $I_0$ set. This result is subsequently applied to obtain sufficient condi…
The statement in the title solves a problem raised by T. Retta. We also present a variation of the result in terms of $[ μ,κ]$-compactness.
A Hausdorff topological semiring is called simple if every non-zero continuous homomorphism into another Hausdorff topological semiring is injective. Classical work by Anzai and Kaplansky implies that any simple compact ring is finite. We generalize this result by proving that every simple compact semiring is finite, i.e., every infinite compact semiring admits a proper non-trivial quotient.
The Hrushovski Property for Compact Special Cube Complexes
本文证明任何紧非正曲率立方复形都可嵌入一个更大的紧非正曲率立方复形，使原复形的组合单射部分局部等距均延拓为整体自同构，并在特殊复形情形下保证嵌入为局部等距。
On Baire property of spaces of compact-valued measurable functions
本文研究可测紧值函数空间$M(X, K)$的Baire性质，证明了在点态收敛拓扑下，对任意可度量化紧空间$K$，该函数空间均为Baire空间，从而解决了通过支撑集拓扑性质刻画函数空间Baire性质的问题。
All spaces are assumed to be separable and metrizable. Our main result is that the statement "For every space $X$, every closed subset of $X$ has the perfect set property if and only if every analytic subset of $X$ has the perfect set property" is equivalent to $\mathfrak{b}>ω_1$ (hence, in particular, it is independent of $\mathsf{ZFC}$). This, together with a theorem of Solecki and an example of Miller, will allow us to determine the status of the statement "For every space $X$, if every $\mathbfΓ$ subset of $X$ has the perfect set property then every $\mathbfΓ'$ subset of $X$ has the perfec…
We study the relationship between generalizations of $P$-spaces and Volterra (weakly Volterra) spaces, that is, spaces where every two dense $G_δ$ have dense (non-empty) intersection. In particular, we prove that every dense and every open, but not every closed subspace of an almost $P$-space is Volterra and that there are Tychonoff non-weakly Volterra weak $P$-spaces. These results should be compared with the fact that every $P$-space is hereditarily Volterra. As a byproduct we obtain an example of a hereditarily Volterra space and a hereditarily Baire space whose product is not weakly Volter…
We show that it is consistent with ZFC that every compact group has a non-Haar-measurable subgroup. In addition, we demonstrate a natural construction, and we conjecture that this construction always produces a non-measurable subgroup of a given compact group. We prove that this is so in the Abelian case.
Every Lindelöf scattered subspace of a $Σ$-product of real lines is $σ$-compact
本文证明第一可数空间$Σ$-积的每个Lindelöf散射子空间是$σ$-紧的，从而回答了Tkachuk提出的问题。
A topological space $Y$ is called a Kempisty space if for any Baire space $X$ every function $f:X\times Y\to\mathbb R$, which is quasi-continuous in the first variable and continuous in the second variable has the Namioka property. Properties of compact Kempisty spaces are studied in this paper. In particular, it is shown that any Valdivia compact is a Kempisty space and the cartesian product of an arbitrary family of compact Kempisty spaces is a Kempisty space.
We prove that any two countable, compact, subsets of $\mathbb{S}^n, n\geq 2$ that are homeomorphic also have homeomorphic complements. Thus any wild subspace like the classical construction of Antoine must contain a Cantor set.
On Relative Invariant Subalgebra Rigidity Property
本文研究了具有特定性质的离散群的相对不变子代数刚性性质，证明了无 torsion 自由的 acylindrically 超几何群满足该性质，并展示了超几何群和高阶半单李群中的不可约晶格的类似性质。
In this paper, we defined two new games - the mildly Menger game and the compact-clopen game. In a zero-dimensional space, the Menger game is equivalent to the mildly Menger game and the compact-open game is equivalent to the compact-clopen game. An example is given for a space on which the mildly Menger game is undetermined. Also we introduced a new game namely K-quasi-component-clopen game and proved that this game is equivalent to the compact-clopen game. Then we proved that if a topological space is a union of countably many quasi-components of compact sets, then TWO has a winning strategy…
In contexts in which the principle of dependent choice may not be available, as toposes or Constructive Set Theory, standard locale theoretic results related to complete regularity may fail to hold. To resolve this difficulty, B. Banaschewski and A. Pultr introduced strongly regular locales. Unfortunately, Banaschewski and Pultr's notion relies on non-constructive set existence principles that hinder its use in Constructive Set Theory. In this article, a fully constructive formulation of strong regularity for locales is introduced by replacing non-constructive set existence with coinductive se…
We present a wide class of reflexive, precompact, non-compact, Abelian topological groups $G$ determined by three requirements. They must have the Baire property, satisfy the \textit{open refinement condition}, and contain no infinite compact subsets. This combination of properties guarantees that all compact subsets of the dual group $G^\wedge$ are finite. We also show that many (non-reflexive) precompact Abelian groups are quotients of reflexive precompact Abelian groups. This includes all precompact almost metrizable groups with the Baire property and their products. Finally, given a compac…
We introduce a notion of productivity (summability) of sequences in a topological group G, parametrized by a given function f : N --> omega+1. The extreme case when f is the function taking constant value omega is closely related to the TAP property, the weaker version of the well-known property NSS. We prove that TAP property coincides with NSS in locally compact groups, omega-bounded abelian groups and countably compact minimal abelian groups. As an application of our results, we provide a negative answer to [13, Question 11.1].
We define the compact universal cover of a compact, metrizable connected space (i.e. a continuum) X to be the inverse limit of all continua that regularly cover X. We show that such covers do indeed form an inverse system with bonding maps that are regular covering maps, and the projection from the inverse limit is a generalized regular covering map in the sense of Berestovskii-Plaut. The inverse limit space is a continuum that is "compactly simply connected" in the sense that its "profinite fundamental group" (the inverse limit of the deck groups of the finite covers) is trivial. We prove a G…
A Wallace semigroup whose every finite power is countably compact
本研究假设存在$\u27e8$个不可比较的选择超滤子，构造出有限幂均可数紧的Wallace半群，证明其可数无穷幂是首个非可数紧的幂次，正面解决了相关公开问题。
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