First Countable Research Articles

Measure-theoretic equicontinuity and rigidity of group actions

Let ( G , X ) be a G-system, which means that X is a compact Hausdorff space and G is an infinite topological group continuously acting on X, and let μ be a G-invariant measure of ( G , X ) . In this paper, we introduce the concepts of rigidity, uniform rigidity and μ-Ω-equicontinuity of ( G , X ) with respect to an infinite sequence Ω of G and the notions of μ-Ω-equicontinuity and μ-Ω-mean-equicontinuity of a function f ∈ L 2 ( μ ) with respect to an infinite sequence Ω of G. Then we give some equivalent conditions for f ∈ L 2 ( μ ) and ( G , X ) to be rigid, respectively. In addition, if G is commutative and X satisfies the first axiom of countability, we present some equivalent conditions for ( G , X ) to be uniformly rigid.

Cardinal inequalities involving the Hausdorff pseudocharacter

We establish several bounds on the cardinality of a topological space involving the Hausdorff pseudocharacter Hpsi (X). This invariant has the property psi _c(X)le Hpsi (X)le chi (X) for a Hausdorff space X. We show the cardinality of a Hausdorff space X is bounded by 2^{pwL_c(X)Hpsi (X)}, where pwL_c(X)le L(X) and pwL_c(X)le c(X). This generalizes results of Bella and Spadaro, as well as Hodel. We show additionally that if X is a Hausdorff linearly Lindelöf space such that Hpsi (X)=omega , then |X|le 2^omega , under the assumption that either 2^{<{mathfrak {c}}}={mathfrak {c}} or {mathfrak {c}}<aleph _omega . The following game-theoretic result is shown: if X is a regular space such that player two has a winning strategy in the game G^{kappa }_1({mathcal {O}}, {mathcal {O}}_D), H psi (X) < kappa and chi (X) le 2^{<kappa }, then |X| le 2^{<kappa }. This improves a result of Aurichi, Bella, and Spadaro. Generalizing a result for first-countable spaces, we demonstrate that if X is a Hausdorff almost discretely Lindelöf space satisfying Hpsi (X)=omega , then |X|le 2^omega under the assumption 2^{<{mathfrak {c}}}={mathfrak {c}}. Finally, we show |X|le 2^{wL(X)Hpsi (X)} if X is a Hausdorff space with a pi -base with elements with compact closures. This is a variation of a result of Bella, Carlson, and Gotchev.

Cardinal estimates involving the weak Lindel\xf6f game

We show that if X is a first-countable Urysohn space where player II has a winning strategy in the game G^{omega _1}_1({mathcal {O}}, {mathcal {O}}_D) (the weak Lindelöf game of length omega _1) then X has cardinality at most continuum. This may be considered a partial answer to an old question of Bell, Ginsburg and Woods. It is also the best result of this kind since there are Hausdorff first-countable spaces of arbitrarily large cardinality where player II has a winning strategy even in the weak Lindelöf game of countable length. We also tackle the problem of finding a bound on the cardinality of a first-countable space where player II has a winning strategy in the game G^{omega _1}_{fin}({mathcal {O}}, {mathcal {O}}_D), providing some partial answers to it. We finish by constructing an example of a compact space where player II does not have a winning strategy in the weak Lindelöf game of length omega _1.

First-countable spaces every quotient of which is orderable

We prove the following Main Theorem: Every Hausdorff quotient image of a first-countable Hausdorff topological space X is a linearly ordered topological space (LOTS) if and only if X is a metrizable space which is the union of a discrete subspace and a compact countable subspace. As a corollary we characterize 1) σ-compact spaces, 2) locally compact spaces, 3) separable spaces every quotient image of which is a LOTS.We examine also several natural examples of non-first-countable Hausdorff topological spaces X such that every quotient image of X is a LOTS.

Cardinal invariants of cellular-Lindelöf spaces

A space X is said to be cellular-Lindelof if for every cellular family $$\mathcal {U}$$ there is a Lindelof subspace L of X which meets every element of $$\mathcal {U}$$ . Cellular-Lindelof spaces generalize both Lindelof spaces and spaces with the countable chain condition. Solving questions of Xuan and Song, we prove that every cellular-Lindelof monotonically normal space is Lindelof and that every cellular-Lindelof space with a regular $$G_\delta $$ -diagonal has cardinality at most $$2^\mathfrak {c}$$ . We also prove that every normal cellular-Lindelof first-countable space has cardinality at most continuum under $$2^{<\mathfrak {c}}=\mathfrak {c}$$ and that every normal cellular-Lindelof space with a $$G_\delta $$ -diagonal of rank 2 has cardinality at most continuum.

The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaces

Abstract This article fits in the context of the approach to topological problems in terms of the underlying convergence space structures, and serves as yet another illustration of the power of the method. More specifically, we spell out convergence-theoretic characterizations of the notions of weak base, weakly first-countable space, semi-metrizable space, and symmetrizable spaces. With the help of the already established similar characterizations of the notions of Frchet-Ursyohn, sequential, and accessibility spaces, we give a simple algebraic proof of a classical result regarding when a symmetrizable (respectively, weakly first-countable, respectively sequential) space is semi-metrizable (respectively first-countable, respectively Fréchet) that clarifies the situation for non-Hausdorff spaces. Using additionally known results on the commutation of the topologizer with product, we obtain simple algebraic proofs of various results of Y. Tanaka on the stability under product of symmetrizability and weak first-countability, and we obtain the same way a new characterization of spaces whose product with every metrizable topology is weakly first-countable, respectively symmetrizable.

Topological spaces with an $\omega^{\omega}$-base

Given a partially ordered set $P$ we study properties of topological spaces $X$ admitting a $P$-base, i.e., an indexed family $(U_\alpha)_{\alpha\in P}$ of subsets of $X\times X$ such that $U_\beta\subset U_\alpha$ for all $\alpha\le\beta$ in $P$ and for every $x\in X$ the family $(U_\alpha[x])_{\alpha\in P}$ of balls $U_\alpha[x]=\{y\in X:(x,y)\in U_\alpha\}$ is a neighborhood base at $x$. A $P$-base $(U_\alpha)_{\alpha\in P}$ for $X$ is called locally uniform if the family of entourages $(U_\alpha U_\alpha^{-1}U_\alpha)_{\alpha\in P}$ remains a $P$-base for $X$. A topological space is first-countable if and only if it has an $\omega$-base. By Moore's Metrization Theorem, a topological space is metrizable if and only if it is a $T_0$-space with a locally uniform $\omega$-base. In the paper we shall study topological spaces possessing a (locally uniform) $\omega^\omega$-base. Our results show that spaces with an $\omega^\omega$-base share some common properties with first countable spaces, in particular, many known upper bounds on the cardinality of first-countable spaces remain true for countably tight $\omega^\omega$-based topological spaces. On the other hand, topological spaces with a locally uniform $\omega^\omega$-base have many properties, typical for generalized metric spaces. Also we study Tychonoff spaces whose universal (pre- or quasi-) uniformity has an $\omega^\omega$-base and show that such spaces are close to being $\sigma$-compact.

Almost fully closed mappings and quasi-$F$-compacta.

The notion of an almost fully closed mapping, which generalizes the definition of a fully closed map, is introduced. It is shown that the composition of almost fully closed mappings of compacta with the first axiom of countability is also almost fully closed, and so is the limit of a countable inverse transfinite sequence of the same mappings. A class of quasi- $F$ - compacta (which contains the class of Fedorchuk compacta ) is defined. For this class, some statements are proved that generalize and strengthen the known theorems on $F$ - compacta .

On the Jayne–Rogers theorem

J. E. Jayne and C. A. Rogers [3] proved that a mapping $${f \colon {X \rightarrow Y}}$$ of an absolute Souslin- $${\mathcal{F}}$$ set X to a metric space Y is $${\mathbf{\Delta}^0_2}$$ -measurable if and only if it is piecewise continuous. We give a similar result for a perfectly paracompact first-countable space X and a regular space Y.

A Study in Grzegorczyk Point-Free Topology Part I: Separation and Grzegorczyk Structures

This is the first, out of two papers, devoted to Andrzej Grzegorczyk’s point-free system of topology from Grzegorczyk (Synthese 12(2–3):228–235, 1960. https://doi.org/10.1007/BF00485101). His system was one of the very first fully fledged axiomatizations of topology based on the notions of region, parthood and separation (the dual notion of connection). Its peculiar and interesting feature is the definition of point, whose intention is to grasp our geometrical intuitions of points as systems of shrinking regions of space. In this part we analyze (quasi-)separation structures and Grzegorczyk structures, and establish their properties which will be useful in the sequel. We prove that in the class of Urysohn spaces with countable chain condition, to every topologically interpreted representative of a point in the sense of Grzegorczyk’s corresponds exactly one point of a space. We also demonstrate that Tychonoff first-countable spaces give rise to complete Grzegorczyk structures. The results established below will be used in the second part devoted to points and topological spaces.

On properties of infimal topology of a map space

We study properties of the infimal topology τinf which is the infimum of the family of all topologies of uniform convergence defined on the set C(X, Y) of continuous maps into a metrizable space Y. One of the main results of the research consists in obtaining necessary and sufficient condition for the topology τinf to have the Frechet–Urysohn property. We also establish necessary and sufficient conditions for coincidence of the topology τinf and a topology of uniform convergence τμ (“attaining” the infimum). We prove that for this coincidence it is sufficient for the topology τinf to satisfy the first axiom of countability.

Multiplication of convex sets in $C(K)$ spaces

Let C(K) denote the Banach algebra of continuous real functions, with the supremum norm, on a compact Hausdorff space K. For two subsets of C(K), one can define their product by pointwise multiplication, just as the Minkowski sum of the sets is defined by pointwise addition. We study this for a special class of closed convex sets, namely closed intervals. Our main interest is in correlations between properties of the product of order intervals in C(K) and properties of the underlying space K. When K is finite, the product of two intervals in C(K) is always an interval. Surprisingly, the converse of this result is true for a wide class of compacta. We show that a first-countable space K is finite whenever it has the property that the product of two nonnegative intervals is closed, or the property that the product of an interval with itself is convex. That some assumption on K is needed, can be seen from the fact that, if K is the Stone–Cech compactification of N, then the product of two intervals in C(K) with continuous boundary functions is always an interval. For any K, it is proved that the product of two positive intervals is always an interval, and that the product of two nonnegative intervals is always convex. Finally, square roots of intervals are investigated, with results of similar type. Mathematics Subject Classification: primary 52A05; secondary 46J10, 46E15.

An internal characterisation of radiality

In this paper, we will investigate how radiality occurs in topological spaces by considering neighbourhood bases generated by nests. We will define a new subclass of radial spaces that contains LOTS, GO-spaces and spaces with well-ordered neighbourhood bases, called the independently-based spaces. We show that first-countable spaces are precisely the independently-based, strongly Fréchet spaces and we give an example of a Fréchet–Urysohn space that is neither independently-based nor strongly Fréchet.

On the cardinality of n-Urysohn and n-Hausdorff spaces

Abstract Two variations of Arhangelskii’s inequality $$\left| X \right| \leqslant 2^{\chi (X) - L(X)}$$ for Hausdorff X [Arhangel’skii A.V., The power of bicompacta with first axiom of countability, Dokl. Akad. Nauk SSSR, 1969, 187, 967–970 (in Russian)] given in [Stavrova D.N., Separation pseudocharacter and the cardinality of topological spaces, Topology Proc., 2000, 25(Summer), 333–343] are extended to the classes with finite Urysohn number or finite Hausdorff number.

Continuous extensions of functions defined on subsets of products

A subset Y of a space X is Gδ-dense if it intersects every nonempty Gδ-set. The Gδ-closure of Y in X is the largest subspace of X in which Y is Gδ-dense.The space X has a regularGδ-diagonal if the diagonal of X is the intersection of countably many regular-closed subsets of X×X.Consider now these results: (a) (N. Noble, 1972 [18]) every Gδ-dense subspace in a product of separable metric spaces is C-embedded; (b) (M. Ulmer, 1970 [22], 1973 [23]) every Σ-product in a product of first-countable spaces is C-embedded; (c) (R. Pol and E. Pol, 1976 [20], also A.V. Arhangelʼskiĭ, 2000 [3]; as corollaries of more general theorems), every dense subset of a product of completely regular, first-countable spaces is C-embedded in its Gδ-closure.The present authorsʼ Theorem 3.10 concerns the continuous extension of functions defined on subsets of product spaces with the κ-box topology. Here is the case κ=ω of Theorem 3.10, which simultaneously generalizes the above-mentioned results. TheoremLet{Xi:i∈I}be a set ofT1-spaces, and let Y be dense in an open subspace ofXI:=∏i∈IXi. Ifχ(qi,Xi)⩽ωfor everyi∈Iand every q in theGδ-closure of Y inXI, then for every regular space Z with a regularGδ-diagonal, every continuous functionf:Y→Zextends continuously over theGδ-closure of Y inXI.Some examples are cited to show that the hypothesis χ(qi,Xi)⩽ω cannot be replaced by the weaker hypothesis ψ(qi,Xi)⩽ω.

ORDER PRESERVING BIJECTIONS OF C+(X)

Let $X$ be a compact Hausdorff space which satisfies the first axiom of countability, and $C_+(X)$ the set of all continuous functions from $X$ to $[0,\infty)$. If $\varphi : C_{+}(X) \rightarrow C_{+}(X)$ is a bijective map which preserves the order in both directions, then there exists a homeomorphism $\omega : X \mathcal{\rightarrow } X$ and for each $x \in X$ a bijective, increasing map $m_{x} : [0,\infty) \mathcal{\rightarrow } [0,\infty)$ such that $\varphi(f)(x) = m_{x}(f(\omega(x)))$, for all $x \in X$ and $f \in C_{+}(X)$.

Lambda-topology versus pointwise topology

AbstractThis paper deals with families of conformal iterated function systems (CIFSs). The space CIFS(X,I) of all CIFSs, with common seed space X and alphabet I, is successively endowed with the topology of pointwise convergence and the so-calledλ-topology. We show just how bad the topology of pointwise convergence is: although the Hausdorff dimension function is continuous on a dense Gδ-set, it is also discontinuous on a dense subset of CIFS(X,I). Moreover, all of the different types of systems (irregular, critically regular, etc.), have empty interior, have the whole space as boundary, and thus are dense in CIFS(X,I), which goes against intuition and conception of a natural topology on CIFS(X,I). We then prove how good the λ-topology is: Roy and Urbański [Regularity properties of Hausdorff dimension in infinite conformal IFSs. Ergod. Th. &amp; Dynam. Sys.25(6) (2005), 1961–1983] have previously pointed out that the Hausdorff dimension function is then continuous everywhere on CIFS(X,I). We go further in this paper. We show that (almost) all of the different types of systems have natural topological properties. We also show that, despite not being metrizable (as it does not satisfy the first axiom of countability), the λ-topology makes the space CIFS(X,I) normal. Moreover, this space has no isolated points. We further prove that the conformal Gibbs measures and invariant Gibbs measures depend continuously on Φ∈CIFS(X,I) and on the parameter t of the potential and pressure functions. However, we demonstrate that the coding map and the closure of the limit set are discontinuous on an important subset of CIFS(X,I).

Joint continuity of K h C-functions with values in moore spaces

We introduce the notion of categorical cliquish mapping and show that, for each K h C-mapping f: X × Y → Z, where X is a topological space, Y is a space with the first axiom of countability, and Z is a Moore space, with categorical-cliquish horizontal y-sections f y , the sets C y (f) are residual G δ-type sets in X for every y ∈ Y.

A topological reflection principle equivalent to Shelah’s strong hypothesis

We notice that Shelah's Strong Hypothesis is equivalent to the following reflection principle: Suppose (X, r) is a first-countable space whose density is a regular cardinal, κ. If every separable subspace of X is of cardinality at most κ, then the cardinality of X is κ.

Nonhomogeneity in products with [formula omitted]-spaces

This note is a contribution to the study of large compact homogeneous spaces (see [5], [7]) and homogeneity properties of products of compact spaces in particular. It is an application of results from [4] and an attempt to draw more attention to these results. Infinite products of zero-dimensional first-countable spaces are homogeneous ([2]), but the homogeneity of certain spaces cannot be improved by taking powers, or even products. Consider the following property of a space X. (*) X × Y is not homogeneous for any compact space Y . This property is shared by (ω1 +1) ([3]) and the ‘topologist’s sine curve’ (Motorov proved it is not a retract of any compact homogeneous space). In [5] Kunen pointed out that it is not known whether (*) holds for all – or any – infinite F-space(s) and proved a weakening of (*) for infinite F-spaces. His theorem implies, for example, that a product of any family of (nontrivial) F-spaces is never homogeneous. Instead of F-spaces we work with a slightly larger class of compact βN-spaces introduced in [1] (see below for definitions). Kunen’s result and its proof remain correct if ‘F-space’ is replaced by ‘βN-space’ everywhere.