The Cech number of Cp(X) when X is an ordinal space

[EN] The Cech number of a space Z, C(Z), is the pseudocharacter of Z in βZ. In this article we obtain, in ZFC and assuming SCH, some upper and lower bounds of the Cech number of spaces Cp(X) of realvalued continuous functions defined on an ordinal space X with the pointwise convergence topologyResearch supported by Fapesp, CONACyT and UNAM.Alas, OT.; Tamariz-Mascarúa, Á. (2008). The Cech number of Cp(X) when X is an ordinal space. Applied General Topology. 9(1):67-76. doi:10.4995/agt.2008.1870.SWORD67769

Ultrafilters and non-Cantor minimal sets in linearly ordered dynamical systems

It is well known that infinite minimal sets for continuous functions on the interval are Cantor sets; that is, compact zero dimensional metrizable sets without isolated points. On the other hand, it was proved in Alcaraz and Sanchis (Bifurcat Chaos 13:1665–1671, 2003) that infinite minimal sets for continuous functions on connected linearly ordered spaces enjoy the same properties as Cantor sets except that they can fail to be metrizable. However, no examples of such subsets have been known. In this note we construct, in ZFC, 2c non-metrizable infinite pairwise nonhomeomorphic minimal sets on compact connected linearly ordered space

Some notes on topological calibers

We show that the definition of caliber given by Engelking in R. Engelking, "General topology", Sigma series in pure mathematics, Heldermann, vol. 6, 1989, which we will call caliber*, differs from the traditional notion of this concept in some cases and agrees in others. For instance, we show that if $\kappa$ is an infinite cardinal with $2^{\kappa}<\aleph_\kappa$ and $cf(\kappa)>\omega$, then there exists a compact Hausdorff space $X$ such that $o(X)=2^{\aleph_\kappa}=|X|$, $\aleph_\kappa$ is a caliber* for $X$ and $\aleph_\kappa$ is not a caliber for $X$. On the other hand, we obtain that if $\lambda$ is an infinite cardinal number, $X$ is a Hausdorff space with $|X|>1$, $\phi\in \{w ,nw\}$, $o(X) = 2^{\phi(X)}$ and $\mu := o\left(X^\lambda\right)$, then the calibers of $X^\lambda$ and the true calibers* (that is, those which are less than or equal to $\mu$) coincide, and are precisely those that have uncountable cofinality

On spaces with star kernel Menger

Given a topological property $\mathcal{P}$, a space $X$ is called star-$\mathcal{P}$ if for any open cover $\mathcal{U}$ of the space $X$, there exists a set $Y\subseteq X$ with property $\mathcal{P}$ such that $St(Y,\mathcal{U})=X$; the set $Y$ is called a star kernel of the cover $\mathcal{U}$. In this paper, we introduce and study spaces with star kernel Menger, that is, star Menger spaces. Some examples are given to show the relationship with some other related properties studied previously, and the behaviour of the star Menger property with respect to subspaces, products, continuous images and preimages are investigated. Additionally, some comments on the star selection theory are given. Particularly, some questions posed by Song within this theory are addressed. Finally, several new properties are introduced as well as some general questions on them are posed

Continuous selections on spaces of continuous functions

summary:For a space $Z$, we denote by \Cal{F}(Z), \Cal{K}(Z) and \Cal{F}_2(Z) the hyperspaces of non-empty closed, compact, and subsets of cardinality $\leq 2$ of $Z$, respectively, with their Vietoris topology. For spaces $X$ and $E$, $C_p(X,E)$ is the space of continuous functions from $X$ to $E$ with its pointwise convergence topology. We analyze in this article when \Cal{F}(Z), \Cal{K}(Z) and \Cal{F}_2(Z) have continuous selections for a space $Z$ of the form $C_p(X,E)$, where $X$ is zero-dimensional and $E$ is a strongly zero-dimensional metrizable space. We prove that $C_p(X,E)$ is weakly orderable if and only if $X$ is separable. Moreover, we obtain that the separability of $X$, the existence of a continuous selection for \Cal{K}(C_p(X,E)), the existence of a continuous selection for \Cal{F}_2(C_p(X,E)) and the weak orderability of $C_p(X,E)$ are equivalent when $X$ is $\Bbb{N}$-compact. Also, we decide in which cases $C_p(X,2)$ and $\beta C_p(X,2)$ are linearly orderable, and when $\beta C_p(X,2)$ is a dyadic space

Generalized linearly ordered spaces and weak pseudocompactness

summary:A space $X$ is {\it truly weakly pseudocompact} if $X$ is either weakly pseudocompact or Lindelöf locally compact. We prove that if $X$ is a generalized linearly ordered space, and either (i) each proper open interval in $X$ is truly weakly pseudocompact, or (ii) $X$ is paracompact and each point of $X$ has a truly weakly pseudocompact neighborhood, then $X$ is truly weakly pseudocompact. We also answer a question about weakly pseudocompact spaces posed by F. Eckertson in [Eck]

Some results and problems about weakly pseudocompact spaces

summary:A space $X$ is {\it truly weakly pseudocompact} if $X$ is either weakly pseudocompact or Lindelöf locally compact. We prove: (1) every locally weakly pseudocompact space is truly weakly pseudocompact if it is either a generalized linearly ordered space, or a proto-metrizable zero-dimensional space with $\chi (x,X)>\omega$ for every $x\in X$; (2) every locally bounded space is truly weakly pseudocompact; (3) for $\omega < \kappa <\alpha$, the $\kappa$-Lindelöfication of a discrete space of cardinality $\alpha$ is weakly pseudocompact if $\kappa = \kappa ^\omega$

Spaces of continuous functions, Σ\Sigma-products and Box Topology

summary:For a Tychonoff space $X$, we will denote by $X_0$ the set of its isolated points and $X_{1}$ will be equal to $X\setminus X_{0}$. The symbol $C(X)$ denotes the space of real-valued continuous functions defined on $X$. $\square\Bbb{R}^{\kappa}$ is the Cartesian product $\Bbb{R}^{\kappa}$ with its box topology, and $C_{\square}(X)$ is $C(X)$ with the topology inherited from $\square\Bbb{R}^{X}$. By $\widehat{C}(X_1)$ we denote the set $\{f\in C(X_1) : f$ can be continuously extended to all of $X\}$. A space $X$ is almost-$\omega$-resolvable if it can be partitioned by a countable family of subsets in such a way that every non-empty open subset of $X$ has a non-empty intersection with the elements of an infinite subcollection of the given partition. We analyze $C_\square (X)$ when $X_0$ is $F_\sigma$ and prove: (1) for every topological space $X$, if $X_{0}$ is $F_{\sigma}$ in $X$, and $\emptyset \ne X_{1}\subset \operatorname{cl}_{X}X_{0}$, then $C_{\square}(X)\cong \square\Bbb{R}^{X_{0}}$; (2) for every space $X$ such that $X_{0}$ is $F_{\sigma}$, $\operatorname{cl}_{X}X_{0}\cap X_{1}\ne \emptyset$, and $X_1 \setminus \operatorname{cl}_X X_0$ is almost-$\omega$-resolvable, then $C_{\square}(X)$ is homeomorphic to a free topological sum of $\leq |\widehat{C}(X_1)|$ copies of $\square\Bbb{R}^{X_{0}}$, and, in this case, $C_{\square}(X) \cong \square\Bbb{R}^{X_{0}}$ if and only if $|\widehat{C}(X_1)|\leq 2^{|X_{0}|}$. We conclude that for a space $X$ such that $X_0$ is $F_\sigma$, $C_\square(X)$ is never normal if $|X_0| >\aleph _0$ [La], and, assuming CH, $C_\square (X)$ is paracompact if $|X_0| = \aleph _0$ [Ru2]. We also analyze $C_\square(X)$ when $|X_1| = 1$ and when $X$ is countably compact, and we scrutinize under what conditions $\square\Bbb{R}^\kappa$ is homeomorphic to some of its ``$\Sigma$-products"; in particular, we prove that $\square\Bbb{R}^\omega$ is homeomorphic to each of its subspaces $\{f \in \square\Bbb{R}^\omega : \{n\in \omega : f(n) = 0\}\in p\}$ for every $p \in \omega^*$, and it is homeomorphic to \{f \in \square\Bbb{R}^\omega : \,\, \forall \,\, \epsilon > 0 \,\, \{n\in \omega : |f(n)| < \epsilon\} \in {\Cal{F}}_0\} where \Cal F_0 is the Fréchet filter on $\omega$