Topology and its Applications最新文献

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.

We provide the first explicit example of a cork of ${\mathrm{CP}}^2\#8\overline{{\mathrm{CP}}^2}$. This result gives the current smallest second Betti number of a standard simply-connected closed 4-manifold for which an explicit cork has been found.

We consider a special class of framed links that arise from the hexatangle. Such links are introduced in [3], where it was also analyzed when the 3-manifold obtained after performing integral Dehn surgery on closed pure 3-braids is $S^3$. In the present paper, we analyze the symmetries of the hexatangle and give a list of Artin n-presentations for the trivial group. These presentations correspond to the double-branched covers of the hexatangle that produce $S^3$ after Dehn surgery. Also, using a result of Birman and Menasco [4], we determine which closed pure 3-braids are hyperbolic.

Beben and Theriault proved a theorem on the homotopy fiber of an extension of a map with respect to a cone attachment, which has produced several applications. We give a short and elementary proof of this theorem.

Given a metric space X, we consider certain families of functions $f:X\to R$ having the hereditary oscillation property HSOP and the hereditary continuous restriction property HCRP on large sets. When X is Polish, among them there are families of Baire measurable functions, $\bar{\mu }$-measurable functions (for a finite nonatomic Borel measure μ on X) and Marczewski measurable functions. We obtain their characterizations using a class of equivalent point-set games. In similar aspects, we study cliquish functions, SZ-functions and countably continuous functions.