Topology and its Applications最新文献

We provide a detailed study of two properties of spaces and pairs of spaces, the surjection property and the ϵ-surjection property, that were recently introduced to characterize the notion of computable type arising from computability theory. For a class of spaces including the finite simplicial complexes, we develop techniques to prove or disprove these properties using homotopy and homology theories, and give applications of these results. In particular, we answer an open question on the computable type property, showing that it is not preserved by taking products. We also observe that computable type is decidable for finite simplicial complexes.

We observe that every zero-dimensional simplicial cochain defines a canonical filtration of a finite simplicial complex and deduce upper estimates for the expected Betti numbers of codimension one random subcomplexes in its support. Moreover, a monotony theorem improves these estimates given any packing of disjoint simplices.

A class of topological spaces is projective (resp., ω-projective) if and only if projective systems of spaces (resp., with a countable cofinal subset of indices) in the class are still in the class. A certain number of classes of Hausdorff spaces are known to be, or not to be, (ω-) projective. We examine classes of spaces that are not necessarily Hausdorff. Sober and compact sober spaces form projective classes, but most classes of locally compact spaces are not even ω-projective. Guided by the fact that the stably compact spaces are exactly the locally compact, strongly sober spaces, and that the strongly sober spaces are exactly the sober, coherent, compact, weakly Hausdorff (in the sense of Keimel and Lawson) spaces, we examine which classes defined by combinations of those properties are projective. Notably, we find that coherent sober spaces, compact coherent sober spaces, as well as (locally) strongly sober spaces, form projective classes.

We show that it is consistent to have regular closed non-clopen copies of $N^{⁎}$ within $N^{⁎}$ and a non-trivial self-map of $N^{⁎}$ even if all autohomeomorphisms of $N^{⁎}$ are trivial.

As an improvement of Fletcher-Lindgren order embedding theorem, we show that every completely regular ordered space X is order embedded in the Tychonoff ordered cube of infinite weight of X. For that purpose, we evaluate the minimal cardinality of continuous functions which represent multi-utility of the (pre)order on X. Moreover, for a topological preordered space X which admits a continuous multi-utility representation (or a completely regular ordered space X) and its compact subspace, similar descriptions by maps to Tychonoff ordered cube are also given.

In [1, Lem. A.4.1], Behrens generalized the classical geometric boundary theorem [10, Thm. 2.3.4]. In this article, we will reformulate [1, Lem. A.4.1] to fix a mistake made by Behrens, and prove it using the language of filtered spectra.

All spaces are assumed to be separable and metrizable. Building on work of van Engelen, Harrington, Michalewski and Ostrovsky, we obtain the following results:

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  Every finite-dimensional analytic space is σ-homogeneous with analytic witnesses,

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  Every finite-dimensional analytic space is σ-homogeneous with pairwise disjoint ${\Delta }_2^1$ witnesses.

We prove the statement formulated in the title of the article. Then we apply it to show that there exists Lindelöf P-groups G and H satisfying $w\left(G\right)=w\left(H\right)=\left|G\right|=\left|H\right|={\aleph }_1$ such that G and H are not locally homeomorphic. This solves Problem 4.4.7 from the book (Arhangel'skii and Tkachenko, 2008 [1]) in the negative. Also, we present two homeomorphic complete Abelian P-groups one of which is ω-narrow and the other is not.

We first obtain some upper bounds on the topological Hausdorff dimensions of fractal squares. As a corollary, we give the formula for this dimension of a special class of fractal squares. Combined with previous results, we also complete the computation of the topological Hausdorff dimensions of fractal squares of order three. Some of these require non-trivial constructions of bases. Our results also shed light on the Lipschitz classification of fractal squares.

Let $C^1\left(M\right)$ be the space of continuously differentiable real-valued functions defined on $[-M,M]$. Here, we address an irremediable flaw found in [4], and show that for the typical element f in $C^1\left(M\right)$, there exists a set $S\subseteq [-M,M]$, both residual and of full measure in $[-M,M]$, such that for any $x\in S$, the trajectory generated by Newton's method using f and x either diverges, converges to a root of f, or generates a Cantor set as its attractor. Whenever the Cantor set is the attractor, the dynamics on the attractor are described by a single type of adding machine, so that the dynamics on all of these attracting Cantor sets are topologically equivalent.