29 pages. This version incorporates the correction of Proposition 3.2 to include an additional assumption (Hausdorff), whose necessity has been pointed out by Alexander Gouberman
In this survey, my aim has been to discuss the use of sequences and countable sets in general topology. In this way I have been led to consider five different classes of topological spaces: first countable spaces, sequential spaces, Frechet spaces, spaces of countable tightness and perfect spaces. We are going to look at how these classes are related, and how well the various properties behave under certain operations, such as taking subspaces, products, and images under proper mappings. Where they are not well behaved we take the opportunity to consider some relevant examples, which are often of special interest. For instance, we examine an example of a Frechet space with unique sequential limits that is not Hausdorff. I asked the question of whether there exists in ZFC an example of a perfectly normal space that does not have countable tightness: such an example was supplied and appears below. In our discussion we shall report two independence theorems, one of which forms the solution to the Moore-Mrowka problem. The results that we prove below include characterisation theorems of sequential spaces and Frechet spaces in terms of appropriate classes of continuous mappings, and the theorem that every perfectly regular countably compact space has countable tightness.
In this paper we define fuzzy sequential topology on a non empty set X which is a collection of fuzzy sequential sets satisfying certain conditions given in its definition and many pleasant properties of a countable number of fuzzy topologies on X associated as components of a FST are investigated.
A.V.Arkhangel'skii asked in 1981 if the variety $\mathfrak V$ of topological groups generated by free topological groups on metrizable spaces coincides with the class of all topological groups. We show that if there exists a real-valued measurable cardinal then the variety $\mathfrak V$ is a proper subclass of the class of all topological groups. A topological group $G$ is called $g$-sequential if for any topological group $H$ any sequentially continuous homomorphism $G\to H$ is continuous. We introduce the concept of a $g$-sequential cardinal and prove that a locally compact group is $g$-sequ…
We endow a topological group $(G, τ)$ with a coarse structure defined by the smallest group ideal $S_τ $ on $G$ containing all converging sequences with their limits and denote the obtained coarse group by $(G, S_τ)$. If $G$ is discrete then $(G, S_τ)$ is a finitary coarse group studding in Geometric Group Theory. The main result: if a topological abelian group $(G, τ)$ contains a non-trivial converging sequence then $asdim \ (G, S_τ)= \infty $.
Let $X$ be a first countable Hausdorff topological group. The limit of a sequence in $X$ defines a function denoted by $lim$ from the set of all convergence sequences to $X$. This definition was modified by Connor and Grosse-Erdmann for real functions by replacing $lim$ with an arbitrary linear functional $G$ defined on a linear subspace of the vector space of all real sequences. Çakallı extended the concept to topological group setting and introduced the concept of $G$-sequential compactness and investigated $G$-sequential continuity and $G$-sequential compactness in topological groups. In th…
A topological group $X$ is called connected if the only subsets which are both open and closed are the whole space $X$ and the null set $\emptyset$. A subset of a topological group is connected if the subspace is connected. We say that a subset $A$ of $X$ is $G$-sequentially connected if the only subsets of $A$ which are both $G$-sequentially open and $G$-sequentially closed, with respect to the relative $G$-sequentially open and $G$-sequentially closed subsets of $A$, are open and closed subsets of $A$ are $A$ and the null set, $\emptyset$. We investigate the impact of changing the definition…
We show that it is consistent to have an uncountable sequential group of intermediate sequential order while no countable such groups exist. This is proved by adding $ω_2$ Cohen reals to a model of $\diamondsuit$.
We develop tools to recognize sequential spaces with large inductive dimension zero. We show the Hawaiian earring group $G$ is 0 dimensional, when endowed with the quotient topology, inherited from the space of based loops with the compact open topology. In particular $G$ is $T_4$ and hence inclusion $G \rightarrow F_M (G)$ is a topological embedding into the free topological group $F_M (G)$ in the sense of Markov.
In this paper, we obtain some results on the relationships between different ideal \linebreak convergence modes namely, $\mathcal{I}^\mathcal{K}$, $\mathcal{I}^{\mathcal{K}^*}$, $\mathcal{I}$, $\mathcal{K}$, $\mathcal{I} \cup \mathcal{K}$ and $(\mathcal{I} \cup \mathcal{K})^*$. We introduce a topological space namely $\mathcal{I}^\mathcal{K}$-sequential space and show that the class of $\mathcal{I}^\mathcal{K}$-sequential spaces contain the sequential spaces. Further $\mathcal{I}^\mathcal{K}$-notions of cluster points and limit points of a function are also introduced here. For a given sequenc…
For the algebraic convergence $λ_{\mathrm{s}}$, which generates the well known sequential topology $τ_s$ on a complete Boolean algebra ${\mathbb B}$, we have $λ_{\mathrm{s}}=λ_{\mathrm{ls}}\cap λ_{\mathrm{li}}$, where the convergences $λ_{\mathrm{ls}}$ and $λ_{\mathrm{li}}$ are defined by $λ_{\mathrm{ls}}(x)=\{ \limsup x\}\!\uparrow$ and $λ_{\mathrm{li}}(x)=\{ \liminf x\}\!\downarrow$ (generalizing the convergence of sequences on the Alexandrov cube and its dual). We consider the minimal topology $\mathcal{O}_{\mathrm{lsi}}$ extending the (unique) sequential topologies $\mathcal{O}_{λ_{\mathrm…
In this paper we have studied some important topological properties and characterization of I^K-convergence of functions which is a common generalization of I*-convergence of functions. We also introduce the idea of I^K*-convergence and I^K-limit points of functions.
The natural duality between "topological" and "regular," both considered as convergence space properties, extends naturally to p-regular convergence spaces, resulting in the new concept of a p-topological convergence space. Taking advantage of this duality, the behavior of p-topological and p-regular convergence spaces is explored, with particular emphasis on the former, since they have not been previously studied. Their study leads to the new notion of a neighborhood operator for filters, which in turn leads to an especially simple characterization of a topology in terms of convergence criter…
In this paper we introduce a weak version of level and epigraph convergence for level functions on topological spaces. In the particular case of topological groups we are able to define convolutions in the set of level functions and show that any such function is the limit in level and epigraph of robust functions.
In this paper we have shown that a double sequence in a topological space satisfies certain conditions which in turn are capable to generate a topology on a non empty set. Also we have used the idea of I-convergence of double sequences to study the idea of I-sequentially compactness [3] in the sense of double sequences.
In this paper we have used the idea of I-convergence of sequences and nets to study certain conditions of convergence in a topological space. It has been shown separately that a class of sequences and a class of nets in a non-empty set X which are respectively called I-convergence class of sequences and I-convergence class of nets satisfying these conditions generate a topology on X. Further we have correlated the classes of I-convergent sequences and nets with respect to these topologies with the given classes which satisfy these conditions.
An open question of Gruenhage asks if all strategically selectively separable spaces are Markov selectively separable, a game-theoretic statement known to hold for countable spaces. As a corollary of a result by Berner and Juh$\acute{a}$sz, we note that the strong version of this statement, where the second player is restricted to selecting single points in the rather than finite subsets, holds for all $T_3$ spaces without isolated points. Continuing this investigation, we also consider games related to selective sequential separability, and demonstrate results analogous to those for selective…
We characterize ultrafilter convergence and ultrafilter compactness in linearly ordered and generalized ordered topological spaces. In such spaces, and for every ultrafilter $D$, the notions of $D$-compactness and of $D$-pseudocompactness are equivalent. Any product of initially $λ$-compact generalized ordered topological spaces is still initially $λ$-compact. On the other hand, preservation under products of certain compactness properties are independent from the usual axioms for set theory.
In this paper, we introduce the notion of topologically Banach contraction mapping defined on an arbitrary topological space X with the help of a continuous function $g:X\times X\rightarrow \mathbb{R}$ and investigate the existence of fixed points of such mapping. Moreover, we introduce two types of mappings defined on a non-empty subset of X and produce sufficient conditions which will ensure the existence of best proximity points for these mappings. Our best proximity point results also extend some existing results from metric spaces or Banach spaces to topological spaces. More precisely, ou…
We prove that it is consistent with ZFC that no sequential topological groups of intermediate sequential orders exist. This shows that the answer to a 1981 question of P.~Nyikos is independent of the standard axioms of set theory. The model constructed also provides consistent answers to several questions of D.~Shakhmatov, S.~Todorčević and Uzcátegui. In particular, we show that it is consistent with ZFC that every countably compact sequential group is Fréchet-Urysohn.
In this article, we consider $\mathcal{I}^\mathcal{K}$-convergence to define a new concept of convergence namely, $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence which generalizes the notion of $\mathcal{S}$-$\mathcal{I}$-convergence introduced by Guevara et al. \cite{GSR20} recently. Some properties of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-convergence of sequences and its relation with compact sets are discussed. In particular, we investigate the relation between semi-compactness and semi-Lindeloffness by introducing the notion of $\mathcal{S}$-$\mathcal{I}^\mathcal{K}$-cluster point of…
We answer a question of S.~Todorčević and C.~Uzcátegui from \cite{TU1} by showing that the only possible sequential orders of sequential analytic groups are $1$ and $ω_1$. Other results on the structure of sequential analytic spaces and their relation to other classes of spaces are given as well. In particular, we provide a full topological classification of sequential analytic groups by showing that all such groups are either metrizable or $k_ω$-spaces, which, together with a result by Zelenyuk, implies that there are exactly $ω_1$ non homeomorphic analytic sequential group topologies.
In this paper, we introduce the notion of $\mathcal{M}$-convergence and $\mathcal{MN}$-convergence structures in posets, which, in some sense, generalise the well-known Scott-convergence and order-convergence structures. As results, we give a necessary and sufficient conditions for each generalised convergence structures being topological. These results then imply the following two well-established results: (1) The Scott-convergence structure in a poset $P$ is topological if and only if $P$ is continuous, and (2) The order-convergence structure in a poset $P$ is topological if and only if $P$ …
We consider precompact sequential and Fréchet group topologies and show that some natural constructions of such topologies always result in metrizable groups answering a question of D.~Dikranjan et al. We show that it is consistent that all sequential precompact topologies on countable groups are Fréchet (or even metrizable). For some classes of groups (for example boolean) extra set-theoretic assumptions may be omitted (although in this case such groups do not have to be metrizable). We also build (using $\diamondsuit$) an example of a countably compact Fréchet group that is not $α_3$ and obt…
We introduce a pointfree theory of convergence on lattices and coframes. A convergence lattice is a lattice $L$ with a monotonic map $\lim_L$ from the lattice of filters on $L$ to $L$, meant to be an abstract version of the map sending every filter of subsets to its set of limits. This construction exhibits the category of convergence spaces as a coreflective subcategory of the opposite of the category of convergence lattices. We extend this construction to coreflections between limit spaces and the opposite of so-called limit lattices and limit coframes, between pretopological convergence spa…
We show that the product of any number of sequentially pseudocompact topological spaces is still sequentially pseudocompact. The definition of sequential pseudocompactness can be given in (at least) two ways: we show their equivalence. Some of the results of the present note already appeared in A. Dow, J. R. Porter, R. M. Stephenson, R. G. Woods, Spaces whose pseudocompact subspaces are closed subsets, Appl. Gen. Topol. 5 (2004), 243-264.
We show that there is a compact topological space carrying a measure which is not a weak* limit of finitely supported measures but is in the sequential closure of the set of such measures. We construct compact spaces with measures of arbitrarily high levels of complexity in this sequential hierarchy. It follows that there is a compact space in which the sequential closure cannot be obtained in countably many steps. However, we show that this is not the case for our spaces where the sequential closure is always obtained in countably many steps.
We prove that each non-metrizable sequential rectifiable space $X$ of countable $cs^*$-character contains a clopen rectifiable submetrizable $k_ω$-subspace $H$ and admits an open disjoint cover by subspaces homeomorphic to clopen subspaces of $H$. This implies that each sequential rectifiable space with countable $cs^*$-character either is metrizable or else is a topological sum of submetrizable $k_ω$-spaces. Consequently, $X$ is submetrizable and paracompact. This answers a question of Lin and Shen posed in 2011.
A space $X$ is called a $k_{R}$-space, if $X$ is Tychonoff and the necessary and sufficient condition for a real-valued function $f$ on $X$ to be continuous is that the restriction of $f$ to each compact subset is continuous. In this paper, we discuss the $k_{R}$-property of products of sequential fans and free Abelian topological groups by applying the $κ$-fan introduced by Banakh. In particular, we prove the following two results: (1) The space $S_{ω_{1}}\times S_{ω_{1}}$ is not a $k_{R}$-space. (2) The space $S_ω\times S_{ω_{1}}$ is a $k_{R}$-space if and only if $S_ω\times S_{ω_{1}}$ is a …
We show that, under suitably general formulations, covering properties, accumulation properties and filter convergence are all equivalent notions. This general correspondence is exemplified in the study of products. Let $X$ be a product of topological spaces. We prove that $X$ is sequentially compact if and only if all subproducts by $\leq \mathfrak s$ factors are sequentially compact. If $\mathfrak s = \mathfrak h$, then $X$ is sequentially compact if and only if all factors are sequentially compact and all but at most $<\mathfrak s$ factors are ultraconnected. We give a topological proof of …
We compare the forcing related properties of a complete Boolean algebra B with the properties of the convergences $λ_s$ (the algebraic convergence) and $λ_{ls}$ on B generalizing the convergence on the Cantor and Aleksandrov cube respectively. In particular we show that $λ_{ls}$ is a topological convergence iff forcing by B does not produce new reals and that $λ_{ls}$ is weakly topological if B satisfies condition $(\hbar)$ (implied by the ${\mathfrak t}$-cc). On the other hand, if $λ_{ls}$ is a weakly topological convergence, then B is a $2^{\mathfrak h}$-cc algebra or in some generic extensi…
A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continous and Borel real-valued functions on Tychonoff spaces, with the topology of pointwise convergence. Our results solve a problem stated by Gartside, Lo, and Marsh.
The omega limit sets plays a fundamental role to construct global attractors for topological semi-dynamical systems with continuous time or discrete time. Therefore, it is important to know when omega limit sets become nonempty compact sets. The purpose of this paper is to understand the mechanism under which a given net of subsets of topological spaces is compact in the asymptotic sense. For this purpose, we introduce the notion of asymptotic compactness for nets of subsets and study the connection with the compactness of the limit sets. In this paper, for a given net of nonempty subsets, we …
In this paper, we study the topological structure of the Hahn field whose elements are functions from the additive abelian group of rational numbers to the real numbers field, with well-ordered support. After reviewing the algebraic and order structures of the Hahn field, we introduce different vector topologies that are induced by families of semi-norms and all of which are weaker than the order or valuation topology. We compare those vector topologies and we identify the weakest one whose properties are similar to those of the weak topology on the Levi-Civita field. In particular, we state a…
