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Let $\Psi \left(\Sigma \right)$ be the Isbell-Mr’owka space associated to the $MAD$-family $\Sigma$. We show that if $G$ is a countable subgroup of the group $𝐒\left(\omega \right)$ of all permutations of $\omega$, then there is a $MAD$-family $\Sigma$ such that every $f\in G$ can be extended to an autohomeomorphism of $\Psi \left(\Sigma \right)$. For a $MAD$-family $\Sigma$, we set $Inv\left(\Sigma \right)=\{f\in 𝐒(\omega ):f[A]\in \Sigma$ for all $A\in \Sigma \}$. It is shown that for every $f\in 𝐒\left(\omega \right)$ there is a $MAD$-family $\Sigma$ such that $f\in Inv\left(\Sigma \right)$. As a consequence of this result we have that there is a $MAD$-family $\Sigma$ such that $n+A\in \Sigma$ whenever $A\in \Sigma$ and $n<\omega$, where $n+A=\{n+a:a\in A\}$ for $n<\omega$. We also notice that there is no $MAD$-family $\Sigma$ such...

Following Kombarov we say that $X$ is $p$-sequential, for $p\in {\alpha }^{*}$, if for every non-closed subset $A$ of $X$ there is $f\in {}^{\alpha }X$ such that $f\left(\alpha \right)\subseteq A$ and $\overline{f}\left(p\right)\in X\setminus A$. This suggests the following definition due to Comfort and Savchenko, independently: $X$ is a FU($p$)-space if for every $A\subseteq X$ and every $x\in A^{-}$ there is a function $f\in {}^{\alpha }A$ such that $\overline{f}\left(p\right)=x$. It is not hard to see that $p\le {}_{RK}q$ ($\le {}_{RK}$ denotes the Rudin–Keisler order) $\iff$ every $p$-sequential space is $q$-sequential $\iff$ every FU($p$)-space is a FU($q$)-space. We generalize the spaces $S_n$ to construct examples of $p$-sequential...

The Katětov ordering of two maximal almost disjoint (MAD) families $𝒜$ and $ℬ$ is defined as follows: We say that $𝒜{\le }_Kℬ$ if there is a function $f:\omega \to \omega$ such that $f^{-1}\left(A\right)\in ℐ\left(ℬ\right)$ for every $A\in ℐ\left(𝒜\right)$. In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called $K$-uniform if for every $X\in ℐ{\left(𝒜\right)}^{+}$, we have that ${𝒜|}_X{\le }_K𝒜$. We prove that CH implies that for every $K$-uniform MAD family $𝒜$ there is a $P$-point $p$ of ${\omega }^{*}$ such that the set of all Rudin-Keisler predecessors of $p$ is dense in the...

For $\varnothing \ne M\subseteq {\omega }^{*}$, we say that $X$ is quasi $M$-compact, if for every $f:\omega \to X$ there is $p\in M$ such that $\overline{f}\left(p\right)\in X$, where $\overline{f}$ is the Stone-Čech extension of $f$. In this context, a space $X$ is countably compact iff $X$ is quasi ${\omega }^{*}$-compact. If $X$ is quasi $M$-compact and $M$ is either finite or countable discrete in ${\omega }^{*}$, then all powers of $X$ are countably compact. Assuming $CH$, we give an example of a countable subset $M\subseteq {\omega }^{*}$ and a quasi $M$-compact space $X$ whose square is not countably compact, and show that in a model of A. Blass and S. Shelah every quasi...

Given a free ultrafilter $p$ on $\mathbb{N}$ and a space $X$, we say that $x\in X$ is the $p$-limit point of a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ in $X$ (in symbols, $x=p$-${lim}_{n\to \infty }{x}_n$) if for every neighborhood $V$ of $x$, $\{n\in \mathbb{N}:x_n\in V\}\in p$. By using $p$-limit points from a suitable metric space, we characterize the selective ultrafilters on $\mathbb{N}$ and the $P$-points of ${\mathbb{N}}^{*}=\beta \left(\mathbb{N}\right)\setminus \mathbb{N}$. In this paper, we only consider dynamical systems $(X,f)$, where $X$ is a compact metric space. For a free ultrafilter $p$ on ${\mathbb{N}}^{*}$, the function $f^p:X\to X$ is defined by $f^p\left(x\right)=p$-${lim}_{n\to \infty }{f}^n\left(x\right)$ for each $x\in X$. These functions are not continuous in general. For a...

It is shown that a space $X$ is $L\left({}^{\mu }p\right)$-Weakly Fréchet-Urysohn for $p\in {\omega }^{*}$ iff it is $L\left({}^{\nu }p\right)$-Weakly Fréchet-Urysohn for arbitrary $\mu ,\nu <{\omega }_1$, where ${}^{\mu }p$ is the $\mu$-th left power of $p$ and $L\left(q\right)=\{{}^{\mu }q:\mu <{\omega }_1\}$ for $q\in {\omega }^{*}$. We also prove that for $p$-compact spaces, $p$-sequentiality and the property of being a $L\left({}^{\nu }p\right)$-Weakly Fréchet-Urysohn space with $\nu <{\omega }_1$, are equivalent; consequently if $X$ is $p$-compact and $\nu <{\omega }_1$, then $X$ is $p$-sequential iff $X$ is ${}^{\nu }p$-sequential (Boldjiev and Malyhin gave, for each $P$-point $p\in {\omega }^{*}$, an example of a compact space $X_p$ which is ${}^{2}p$-Fréchet-Urysohn and it is...

We introduce the properties of a space to be strictly $WFU\left(M\right)$ or strictly $SFU\left(M\right)$, where $\varnothing \ne M\subset {\omega }^{*}$, and we analyze them and other generalizations of $p$-sequentiality ($p\in {\omega }^{*}$) in Function Spaces, such as Kombarov’s weakly and strongly $M$-sequentiality, and Kocinac’s $WFU\left(M\right)$ and $SFU\left(M\right)$-properties. We characterize these in $C_{\pi }\left(X\right)$ in terms of cover-properties in $X$; and we prove that weak $M$-sequentiality is equivalent to $WFU\left(L\right(M\left)\right)$-property, where $L\left(M\right)=\{{}^{\lambda }p:\lambda <{\omega }_1$ and $p\in M\}$, in the class of spaces which are $p$-compact for every $p\in M\subset {\omega }^{*}$; and that $C_{\pi }\left(X\right)$ is a $WFU\left(L\right(M\left)\right)$-space iff $X$ satisfies...

Let $W$ be the subspace of ${\mathbb{N}}^{*}$ consisting of all weak $P$-points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$-pseudocompact space for all $p\in {\mathbb{N}}^{*}$.

In this paper, we deal with the product of spaces which are either $𝒢$-spaces or ${𝒢}_p$-spaces, for some $p\in {\omega }^{*}$. These spaces are defined in terms of a two-person infinite game over a topological space. All countably compact spaces are $𝒢$-spaces, and every ${𝒢}_p$-space is a $𝒢$-space, for every $p\in {\omega }^{*}$. We prove that if $\{X_{\mu }:\mu <{\omega }_1\}$ is a set of spaces whose product $X={\prod }_{\mu <{\omega }_1}{X}_{\mu }$ is a $𝒢$-space, then there is $A\in {\left[{\omega }_1\right]}^{\le \omega }$ such that $X_{\mu }$ is countably compact for every $\mu \in {\omega }_1\setminus A$. As a consequence, $X^{{\omega }_1}$ is a $𝒢$-space iff $X^{{\omega }_1}$ is countably compact, and if $X^{2^{𝔠}}$ is a $𝒢$-space, then all...

For a free ultrafilter $p$ on $\mathbb{N}$, the concepts of strong pseudocompactness, strong $p$-pseudocompactness and pseudo-$\omega$-boundedness were introduced in [Angoa J., Ortiz-Castillo Y.F., Tamariz-Mascarúa A., Ultrafilters and properties related to compactness, Topology Proc. 43 (2014), 183–200] and [García-Ferreira S., Ortiz-Castillo Y.F., Strong pseudocompact properties of certain subspaces of ${\mathbb{N}}^{*}$, submitted]. These properties in a space $X$ characterize the pseudocompactness of the hyperspace $𝒦\left(X\right)$ of compact subsets...

We consider discrete dynamical systems whose phase spaces are compact metrizable countable spaces. In the first part of the article, we study some properties that guarantee the continuity of all functions of the corresponding Ellis semigroup. For instance, if every accumulation point of $X$ is fixed, we give a necessary and sufficient condition on a point $a\in X^{\text{'}}$ in order that all functions of the Ellis semigroup $E(X,f)$ be continuous at the given point $a$. In the second part, we consider transitive dynamical...

Following Malykhin, we say that a space $X$ is if $X$ contains a family $𝒟$ of dense subsets such that $\left|𝒟\right|>\Delta \left(X\right)$ and the intersection of every two elements of $𝒟$ is nowhere dense, where $\Delta \left(X\right)=min\left\{\right|U|:U$ is a nonempty open subset of $X\}$ is the of $X$. We show that, for every cardinal $\kappa$, there is a compact extraresolvable space of size and dispersion character $2^{\kappa }$. In connection with some cardinal inequalities, we prove the equivalence of the following statements: 1) $2^{\kappa }<2^{{\kappa }^{+}}$, 2) ${\left({\kappa }^{+}\right)}^{\kappa }$ is extraresolvable and 3) $A{\left({\kappa }^{+}\right)}^{\kappa }$ is extraresolvable, where $A\left({\kappa }^{+}\right)$...

For a cardinal $\alpha$, we say that a subset $B$ of a space $X$ is $C_{\alpha }$-compact in $X$ if for every continuous function $fX\to {ℝ}^{\alpha }$, $f\left[B\right]$ is a compact subset of ${ℝ}^{\alpha }$. If $B$ is a $C$-compact subset of a space $X$, then $\rho (B,X)$ denotes the degree of $C_{\alpha }$-compactness of $B$ in $X$. A space $X$ is called $\alpha$-pseudocompact if $X$ is $C_{\alpha }$-compact into itself. For each cardinal $\alpha$, we give an example of an $\alpha$-pseudocompact space $X$ such that $X\times X$ is not pseudocompact: this answers a question posed by T. Retta in “Some cardinal generalizations of pseudocompactness”...