证明了第一可数仿紧σ-空间中$\Ind$维数与$\dim$维数的一致性。这正面回答了A. V. Arkhangel'skii的问题：对于具有可数网络的第一可数空间，维数$\ind X$、$\Ind X$和$\dim X$是否相等。
英文摘要：
The coincidence of the $\Ind$ and $\dim$ dimensions for first countable paracompact $σ$-spaces is proved. This gives a positive answer to A. V. Arkhangel'skii's question of whether the dimensions $\ind X$, $\Ind X$, and $\dim X$ are equal for first countable spaces with a countable network.
For a $T_0$ space $X$, let $\mk (X)$ be the poset of all compact saturated sets of $X$ with the reverse inclusion order. The space $X$ is said to have property Q if for any $K_1, K_2\in \mk (X)$, $K_2\ll K_1$ in $\mk (X)$ if{}f $K_2\subseteq \ii~\!K_1$. In this paper, we give several connections among the well-filteredness of $X$, the sobriety of $X$, the local compactness of $X$, the core compactness of $X$, the property Q of $X$, the coincidence of the upper Vietoris topology and Scott topology on $\mk (X)$, and the continuity of $x\mapsto\ua x : X \longrightarrow Σ~\!\! \mk (X)$ (where $Σ~\…
Two selection games from the literature, $G_c(\mathcal O,\mathcal O)$ and $G_1(\mathcal O_{zd},\mathcal O)$, are known to characterize countable dimension among certain spaces. This paper studies their perfect- and limited-information strategies, and investigates issues related to non-equivalent characterizations of zero-dimensionality for spaces that are not both separable and metrizable. To relate results on zero-dimensional and finite-dimensional spaces, a generalization of Telgársky's proof that the point-open and finite-open games are equivalent is demonstrated.
We answer several questions of V. Tkačuk from [Point-countable $π$-bases in first countable and similar spaces, Fund. Math. 186 (2005), pp. 55--69.] by showing that (1) there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable $π$-base (in fact, the order of any $π$-base of the space is at least $\aleph_ω$); (2) if there is a $κ$-Suslin line then there is a first countable GO space of cardinality $κ^+$ in which the order of any $π$-base is at least $κ$; (3) it is consistent to have a first countable, hereditarily Lindel\" of regular space having uncount…
Building on work of Baldwin and Beaudoin, assuming Martin's Axiom, we construct a zero-dimensional separable metrizable space $X$ such that $X$ is countable dense homogeneous while $X^2$ is not. It follows from results of Hrušák and Zamora Avilés that such a space $X$ cannot be Borel. Furthermore, $X$ can be made homogeneous and completely Baire as well.
We show in detail that every compact countable subset of a metric space is homeomorphic to a countable ordinal number, which extends a result given by Mazurkiewicz and Sierpinski for finite-dimensional Euclidean spaces. In order to achieve this goal, we use Transfinite Induction to construct a specific homeomorphism. In addition, we prove that for all metric space $(E,d)$, the cardinality of the set of all the equivalence classes $\mathscr{K}_E$, up to homeomorphisms, of compact countable subsets of $E$ is less than or equal to $\aleph_1$, i.e. $|\mathscr{K}_E| \le \aleph_1$. We also show that…
We show that, for a coanalytic subspace $X$ of $2^ω$, the countable dense homogeneity of $X^ω$ is equivalent to $X$ being Polish. This strengthens a result of Hrušák and Zamora Avilés. Then, inspired by results of Hernández-Gutiérrez, Hrušák and van Mill, using a technique of Medvedev, we construct a non-Polish subspace $X$ of $2^ω$ such that $X^ω$ is countable dense homogeneous. This gives the first $\mathsf{ZFC}$ answer to a question of Hrušák and Zamora Avilés. Furthermore, since our example is consistently analytic, the equivalence result mentioned above is sharp. Our results also answer a…
Let $X$ be a zero-dimensional space and $C_c(X)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_c^K(X)$ (resp., $C_{c}^ψ(X)$) the set of all functions in $C_c(X)$ with compact (resp., pseudocompact) support. First, we observe that $C_c^{K}(X)=O_c^{β_0X\setminus X}$ (resp., $C_c^ψ(X)=M_c^{β_0X\setminus \upsilon_0X}$). This implies that for an $\Bbb{N}$-compact space $X$, the intersection of all free maximal ideals in $C_c(X)$ equals to $C_c^K(X)$, i.e., $M_c^{β_0X\setminus X}=C_c^K(X)$. Afterwards, by applying methods of functiona…
We study the coincidence theory of maps between two manifolds of the same dimension from an axiomatic viewpoint. First we look at coincidences of maps between manifolds where one of the maps is orientation true, and give a set of axioms such that characterizes the local index (which is an integer valued function). Then we consider coincidence theory for arbitrary pairs of maps between two manifolds. Similarly we provide a set of axioms which characterize the local index, which in this case is a function with values in $\Z\oplus \Z_2$. We also show in each setting that the group of values for t…
For a Hausdorff zero-dimensional topological space $X$ and a totally ordered field $F$ with interval topology, let $C_c(X,F)$ be the ring of all $F-$valued continuous functions on $X$ with countable range. It is proved that if $F$ is either an uncountable field or countable subfield of $\mathbb{R}$, then the structure space of $C_c(X,F)$ is $β_0X$, the Banaschewski Compactification of $X$. The ideals $\{O^{p,F}_c:p\in β_0X\}$ in $C_c(X,F)$ are introduced as modified countable analogue of the ideals $\{O^p:p\inβX\}$ in $C(X)$. It is realized that $C_c(X,F)\cap C_K(X,F)=\bigcap_{p\inβ_0X\texttt{…
For $f,g:X\longrightarrow X$ continuous and commuting maps of a Hausdorff space, we investigate various conditions on $X$ and on the pair $(f,g)$ which provide existence of a coincidence value. We introduce generalized notions of the coincidence value property and use this added flexibility to determine how various coincidence properties of $X$ are related to group actions on $X$ and to coincidence properties of associated fibre bundles and adjunction spaces. We also present a sheaf theoretical approach to obtaining information concerning coincidence values through construction of an "almost c…
The Continuum Hypothesis implies an Erdös-Sierpiński like duality between the ideal of first category subsets of $\reals^{\naturals}$, and the ideal of countable dimensional subsets of $\reals^{\naturals}$. The algebraic sum of a Hurewicz subset - a dimension theoretic analogue of Sierpinski sets and Lusin sets - of $\reals^{\naturals}$ with any compactly countable dimensional subset of $\reals^{\naturals}$ has first category.
We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zero-dimensional Borel CDH spaces. We also show that for a Borel X subset of 2^omega the following are equivalent: (1) X is G_delta in 2^omega, (2) X^omega is CDH and (3) X^omega is homeomorphic to 2^omega or to omega^omega. Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. In particular, modulo large cardinal assum…
Michael asked whether every productively Lindelöf space is powerfully Lindelöf. Building of work of Alster and De la Vega, assuming the Continuum Hypothesis, we show that every productively Lindelöf space of countable tightness is powerfully Lindelöf. This strengthens a result of Tall and Tsaban. The same methods also yield new proofs of results of Arkhangel'skii and Buzyakova. Furthermore, assuming the Continuum Hypothesis, we show that a productively Lindelöf space $X$ is powerfully Lindelöf if every open cover of $X^ω$ admits a point-continuum refinement consisting of basic open sets. This …
We consider the category of all locally Lipschitz contractible metric spaces and all locally Lipschitz maps, which is a wide class of metric spaces, including all finite dimensional Alexandrov spaces and all CAT spaces. We also consider the chain complex of normal currents with compact support in a metric space in the sense of Ambrosio and Kirchheim. In the present paper, its homology is proved to be a homotopy invariant on the category. To prove this result, we define a new topology on a space of Lipschitz maps between arbitrary metric spaces. This topology is proved to coincide with the usua…
