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We give sufficient conditions when a topological inverse $λ$-polycyclic monoid $P_λ$ is absolutely $H$-closed in the class of topological inverse semigroups. Also, for every infinite cardinal $λ$ we construct the coarsest semigroup inverse topology $τ_{mi}$ on $P_λ$ and give an example of a topological inverse monoid S which contains the polycyclic monoid $P_2$ as a dense discrete subsemigroup.
We provide a complete classification of the possible cofinal structures of the families of precompact (totally bounded) sets in general metric spaces, and compact sets in general complete metric spaces. Using this classification, we classify the cofinal structure of local bases in the groups $\C(X,\bbR)$ of continuous real-valued functions on complete metric spaces $X$, with respect to the compact-open topology.
We perform a geometric study of the equilibrium locus of the flow that models the diffusion process over a circular network of cells. We prove that when considering the set of all possible values of the parameters, the equilibrium locus is a smooth manifold with corners, while for a given value of the parameters, it is an embedded smooth and connected curve. For different values of the parameters, the curves are all isomorphic. Moreover, we show how to build a homotopy between different curves obtained for different values of the parameter set. This procedure allows the efficient computation of the equilibrium point for each value of some first integral of the system. This point would have been otherwise difficult to be computed for higher dimensions. We illustrate this construction by some numerical experiments. Eventually, we show that when considering the parameters as inputs, one can easily bring the system asymptotically to any equilibrium point in the reachable set, which we also easily characterize.
We answer a question of O. Kalenda and J. Spurný and give an example of a completely regular hereditarily Baire space $X$ and a Baire-one function $f:X\to [0,1]$ which can not be extended to a Baire-one function on $βX$.
We study products of general topological spaces with Menger's covering property, and its refinements based on filters and semifilters. To this end, we extend the projection method from the classic real line topology to the Michael topology. Among other results, we prove that, assuming \CH{}, every productively Lindelöf space is productively Menger, and every productively Menger space is productively Hurewicz. None of these implications is reversible.
We prove that if $H$ is a topological group such that all closed subgroups of $H$ are separable, then the product $G\times H$ has the same property for every separable compact group $G$. Let $c$ be the cardinality of the continuum. Assuming $2^{ω_1} = c$, we show that there exist: (1) pseudocompact topological abelian groups $G$ and $H$ such that all closed subgroups of $G$ and $H$ are separable, but the product $G\times H$ contains a closed non-separable $σ$-compact subgroup; (2) pseudocomplete locally convex vector spaces $K$ and $L$ such that all closed vector subspaces of $K$ and $L$ are separable, but the product $K\times L$ contains a closed non-separable $σ$-compact vector subspace.
A topological space $L$ is called a linear ordered topological space (LOTS) whenever there is a linear order $\leq$ on $L$ such that the topology on $L$ is generated by the open sets of the form $(a, b)$ with $a < b$ and $a, b \in L \cup \{ -\infty, +\infty \}$. A topological space $X$ is called a generalized ordered space (GO-space) whenever $X$ is topologically embeddable in a LOTS. Main Theorem: Let $X$ be a Hausdorff topological space. Assume that any continuous image of $X$ is a GO-space. Then $X$ is homeomorphic to a countable successor ordinal (with the order topology). The converse trivially holds.
Characterizations of skeletally Dugundji spaces and Dugundji spaces are given in terms of club collections, consisting of countable families of co-zero sets. For example, a Tychonoff space $X$ is skeletally Dugundji if and only if there exists an additive $c$-club on $X$. Dugundji spaces are characterized by the existence of additive $d$-clubs.
We introduce and investigate the class of skeletally Dugundji spaces as a skeletal analogue of Dugundji space. The main result states that the following conditions are equivalent for a given space $X$: (i) $X$ is skeletally Dugundji; (ii) Every compactification of $X$ is co-absolute to a Dugundji space; (iii) Every $C^*$-embedding of the absolute $p(X)$ in another space is strongly $π$-regular; (iv) $X$ has a multiplicative lattice in the sense of Shchepin \cite{s76} consisting of skeletal maps.
We examine the class of spaces in which the second player has a winning strategy in the open--open game. We show that this spaces are not universally Kuratowski-Ulam. We also show that the games G and G7 introduced by P. Daniels, K. Kunen, H. Zhou [Fund. Math. 145 (1994), no. 3, 205--220] are not equivalent.
The aim of this note is to investigate the open-open game of uncountable length. We introduce a cardinal number $μ(X)$, which says how long the Player I has to play to ensure a victory. It is proved that $\su(X)\leqμ(X)\leq\su(X)^+$. We also introduce the class $\mathcal C_κ$ of topological spaces that can be represented as the inverse limit of $κ$-complete system $\{X_σ,π^σ_ρ,Σ\}$ with $\w(X_σ)\leqκ$ and skeletal bonding maps. It is shown that product of spaces which belong to $\mathcal C_κ$ also belongs to this class and $μ(X)\leqκ$ whenever $X\in\mathcal C_κ$ .
We consider a version of the open-open game, indicating its connections with universally Kuratowski-Ulam spaces. We show that: Every I-favorable space is universally Kuratowski-Ulam, (Theorem 8); If a compact space Y is I-favorable, then the hyperspace exp(Y) with the Vietoris topology is I-favorable, and hence universally Kuratowski-Ulam, (Theorems 6 and 9). Notions of uK-U and uK-U* spaces are compared.
Given an abstract simplicial complex G, the connection graph G' of G has as vertex set the faces of the complex and connects two if they intersect. If A is the adjacency matrix of that connection graph, we prove that the Fredholm characteristic det(1+A) takes values in {-1,1} and is equal to the Fermi characteristic, which is the product of the w(x), where w(x)=(-1)^dim(x). The Fredholm characteristic is a special value of the Bowen-Lanford zeta function and has various combinatorial interpretations. The unimodularity theorem proven here shows that it is a cousin of the Euler characteristic as the later is the sum of the w(x). Unimodularity implies that the matrix 1+A has an inverse which takes integer values. Experiments suggest the conjecture that the range of the Green function values, the union of the entries of the inverse of 1+A form a combinatorial invariant of the simplicial complex and do not change under Barycentric or edge refinements.
In this short note, a topos - called the topos of the connectivity space - is associated with every such space.
This issue surveys some of the activities in the field since the previous issue, and announces a conference fully dedicated to the topic of selection principles.
An $H$-closed quasitopological group is a Hausdorff quasitopological group which is contained in each Hausdorff quasitopological group as a closed subspace. We obtained a sufficient condition for a quasitopological group to be $H$-closed, which allowed us to solve a problem by Arhangel'skii and Choban and to show that a topological group $G$ is $H$-closed in the class of quasitopological groups if and only if $G$ is Ra\vıkov-complete. Also we present examples of non-compact quasitopological groups whose topological spaces are $H$-closed.
A topological space $X$ is called almost discretely Lindelöf if every discrete set $D \subset X$ is included in a Lindelöf subspace of $X$. We say that the space $X$ is {\em $μ$-sequential} if for every non-closed set $A \subset X$ there is a sequence of length $\le μ$ in $A$ that converges to a point which is not in $A$. With the help of a technical theorem that involves elementary submodels, we establish the following two results concerning such spaces. (1) For every almost discretely Lindelöf $T_3$ space $X$ we have $|X| \le 2^{χ(X)}$. (2) If $X$ is a $μ$-sequential $T_2$ space of pseudocharacter $ψ(X) \le 2^μ$ and for every free set $D \subset X$ we have $L(\overline{D}) \le μ$, then $|X| \le 2^μ$. The case $χ(X) = ω$ of (1) provides a solution to Problem 4.5 from "I. Juhász, V. Tkachuk, and R. Wilson, Weakly linearly Lindelöf monotonically normal spaces are Lindelöf", while the case $μ= ω$ of (2) is a partial improvement on the main result of "A.V. Archangel'skii and R.Z. Buzyakova, On some properties of linearly Lindelöf spaces".
We show that for arbitrary linearly ordered set $X$ any bounded family of (not necessarily, continuous) real valued functions on $X$ with bounded total variation does not contain independent sequences. We obtain generalized Helly's sequential compactness type theorems. One of the theorems asserts that for every compact metric space $(Y,d)$ the compact space $BV_r(X,Y)$ of all functions $X \to Y$ with variation $\leq r$ is sequentially compact in the pointwise topology. Another Helly type theorem shows that the compact space $M_+(X,Y)$ of all order preserving maps $X \to Y$ is sequentially compact where $Y$ is a compact metrizable partially ordered space in the sense of Nachbin.
We describe the Lelek fan, a smooth fan whose set of end-points is dense, and the Poulsen simplex, a Choquet simplex whose set of extreme points is dense, as Fraïssé limits in certain natural categories of embeddings and projections. As an application we give a short proof of their uniqueness, universality, and almost homogeneity. We further show that for every two countable dense subsets of end-points of the Lelek fan there exists an auto-homeomorphism of the fan mapping one set onto the other. This improves a result of Kawamura, Oversteegen, and Tymchatyn from 1996.
Two of the natural topologies for infinite graphs with edge-ends are Etop and Itop. In this paper, we study and characterize them. We show that Itop can be constructed by inverse limits of inverse systems of graphs with finitely many vertices. Furthermore, as an application of the inverse limit approach, we construct a topological spanning tree in Itop
Known and new results on free Boolean topological groups are collected. An account of properties which these groups share with free or free Abelian topological groups and properties specific of free Boolean groups is given. Special emphasis is placed on the application of set-theoretic methods to the study of Boolean topological groups.
Suppose that $\{T_{a}:a\in G\}$ is a group of uniformly $L$-Lipschitzian mappings with bounded orbits $\left\{T_{a}x:a\in G\right\}$ acting on a hyperconvex metric space $M$. We show that if $L<\sqrt{2}$, then the set of common fixed points $Fix \, G$ is a nonempty Hölder continuous retract of $M$. As a consequence, it follows that all surjective isometries acting on a bounded hyperconvex space have a common fixed point. A fixed point theorem for $L$-Lipschitzian involutions and some generalizations to the case of $λ$-hyperconvex spaces are also given.
We provide a very short elementary proof that every bounded separable metric group embeds into a monothetic bounded metric group, in such a way that the result of Morris and Pestov that every separable abelian topological group embeds into a monothetic group is an immediate corollary. We show that the boundedness assumption is essential.
Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graph on the IFS has been established recently. In the paper, we introduce a notion of simple augmented tree which is a Gromov hyperbolic graph. By generalizing a combinatorial device of rearrangeable matrix, we show that there exists a near-isometry between the simple augmented tree and the symbolic space of the IFS, so that their hyperbolic boundaries are Lipschitz equivalent. We then apply this to consider the Lipschitz equivalence of self-similar sets with or without assuming the open set condition. Moreover, we also provide a criterion for a self-similar set to be a Cantor-type set which completely answers an open question raised in \cite{LaLu13}. Our study extends the previous works.
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