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For a space $Z$, we denote by $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ the hyperspaces of non-empty closed, compact, and subsets of cardinality $\le 2$ 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 $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ 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...

A dense-in-itself space $X$ is called if the space of real continuous functions on $X$ with its box topology, $C_{\square }\left(X\right)$, is a discrete space. A space $X$ is called provided that $X$ is the union of a countable increasing family of subsets each of them with an empty interior. We analyze these classes of spaces by determining their relations with $\kappa$-resolvable and almost resolvable spaces. We prove that every almost-$\omega$-resolvable space is $C_{\square }$-discrete, and that these classes coincide in the realm of completely regular...

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 $C\left(X\right)$ denotes the space of real-valued continuous functions defined on $X$. $\square {ℝ}^{\kappa }$ is the Cartesian product ${ℝ}^{\kappa }$ with its box topology, and $C_{\square }\left(X\right)$ is $C\left(X\right)$ with the topology inherited from $\square {ℝ}^X$. By $\widehat{C}\left(X_1\right)$ we denote the set $\{f\in C\left(X_1\right):f$ 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...

Let $X$ be a Hausdorff space and let $\mathscr{H}$ be one of the hyperspaces $CL\left(X\right)$, $𝒦\left(X\right)$, $ℱ\left(X\right)$ or ${ℱ}_n\left(X\right)$ ($n$ a positive integer) with the Vietoris topology. We study the following disconnectedness properties for $\mathscr{H}$: extremal disconnectedness, being a $F^{\text{'}}$-space, $P$-space or weak $P$-space and hereditary disconnectedness. Our main result states: if $X$ is Hausdorff and $F\subset X$ is a closed subset such that (a) both $F$ and $X-F$ are totally disconnected, (b) the quotient $X/F$ is hereditarily disconnected, then $𝒦\left(X\right)$ is hereditarily disconnected. We also...

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...

A space $X$ is 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].

A space $X$ is 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 $\kappa ={\kappa }^{\omega }$.

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...

We prove that, assuming MA, every crowded $T_0$ space $X$ is $\omega$-resolvable if it satisfies one of the following properties: (1) it contains a $\pi$-network of cardinality $<𝔠$ constituted by infinite sets, (2) $\chi \left(X\right)<𝔠$, (3) $X$ is a $T_2$ Baire space and $c\left(X\right)\le {\aleph }_0$ and (4) $X$ is a $T_1$ Baire space and has a network $𝒩$ with cardinality $<𝔠$ and such that the collection of the finite elements in it constitutes a $\sigma$-locally finite family. Furthermore, we prove that the existence of a $T_1$ Baire irresolvable space is equivalent to the existence of...

We continue the study of almost-$\omega$-resolvable spaces beginning in A. Tamariz-Mascar’ua, H. Villegas-Rodr’ıguez, , Comment. Math. Univ. Carolin. (2002), no. 4, 687–705. We prove in ZFC: (1) every crowded $T_0$ space with countable tightness and every $T_1$ space with $\pi$-weight $\le {\aleph }_1$ is hereditarily almost-$\omega$-resolvable, (2) every crowded paracompact $T_2$ space which is the closed preimage of a crowded Fréchet $T_2$ space in such a way that the crowded part of each fiber is $\omega$-resolvable, has this property too, and (3)...

Given a Tychonoff space $X$, a base $\alpha$ for an ideal on $X$ is called pseudouniform if any sequence of real-valued continuous functions which converges in the topology of uniform convergence on $\alpha$ converges uniformly to the same limit. This paper focuses on pseudouniform bases for ideals with particular emphasis on the ideal of compact subsets and the ideal of all countable subsets of the ground space.

For a topological property $P$, we say that a space $X$ is star $P$ if for every open cover $𝒰$ of the space $X$ there exists $A\subset X$ such that $st(A,𝒰)=X$. We consider space with star countable extent establishing the relations between the star countable extent property and the properties star Lindelöf and feebly Lindelöf. We describe some classes of spaces in which the star countable extent property is equivalent to either the Lindelöf property or separability. An example is given of a Tychonoff star Lindelöf space with...