Recent zbMATH articles in MSC 54G20 https://zbmath.org/atom/cc/54G20 2021-05-28T16:06:00+00:00 Werkzeug For every space $$X$$, either $$C_{p}C_{p}(X)$$ or $$C_{p}C_{p}C_{p}(X)$$ is $$\psi$$-separable. https://zbmath.org/1459.54014 2021-05-28T16:06:00+00:00 "Tkachuk, V. V." https://zbmath.org/authors/?q=ai:tkachuk.vladimir-v For every space $$X$$, either $$C_{p}C_{p}(X)$$ or $$C_{p}C_{p}C_{p}(X)$$ is $$\psi$$-separable, cf. [\textit{V. V. Tkachuk}, Topology Appl. 284, Article ID 107362, 9 p. (2020; Zbl 07286521)]. A space with a dense subspace of countable pseudocharacter is called $$\psi$$-separable. In this paper the following results are proved: \textbf{3.1 Theorem}. Suppose that $$X$$ is a space and consider the following conditions for the cardinal $$\kappa=\min\{i\omega(X), d(X)\}:$$ $$(\Omega_{0})$$ for any $$U\in\tau(X)$$, if $$d(X\backslash U)<\kappa$$ then $$i\omega(U)=i\omega(X)$$; $$(\Omega_{1})$$ $$i\omega(X)<d(X)$$ and hence $$\kappa=i\omega(X)$$; $$(\Omega_{2})$$ there exists a discrete subset $$D\subset X\times X$$ such that $$|D|\geq\kappa$$. Then, (a) $$(\Omega_{0})$$ implies $$(\Omega_{2})$$ in ZFC. (b) $$(\Omega_{1})$$ implies $$(\Omega_{2})$$ under GCH. \textbf{3.3 Theorem}. If $$X$$ is a space such that $$d(X)\leq i\omega(X)$$, then $$C_{p,2n}(X)$$ has a dense $$F_{\sigma}$$-discrete subspace for all $$n\in\mathbb{N}$$. \textbf{3.4 Theorem}. If $$X$$ is a space such that $$i\omega(X)\leq d(X)$$, then $$C_{p,2n+1}(X)$$ has a dense $$F_{\sigma}$$-discrete subspace for all $$n\in\mathbb{N}$$. \textbf{3.14 Theorem}. If the Generalized Continuum Hypothesis (GCH) holds, then for any space $$X$$, either $$C_{p,2n}(X)$$ has a dense $$F_{\sigma}$$-discrete subspace for every $$n\in\mathbb{N}$$ or $$C_{p,2n-1}(X)$$ has a dense $$F_{\sigma}$$-discrete subspace for each $$n\in\mathbb{N}$$. \textbf{4.2 Theorem}. If $$X$$ is an infinite metrizable space, then $$C_{p}(X)$$ has a uniformly dense subspace of countable pseudocharacter if and only if $$2^{\omega(X)}=\omega(X)^{\omega}$$. \textbf{4.4 Theorem}. If $$X$$ is an infinite Corson compact space, then for any $$n\in\mathbb{N}$$, there exists a simple closed subspace $$T\subset C_{p,n}(X)$$ such that $$\lvert T\rvert = \omega(X)$$. \textbf{4.6 Theorem}. Given an infinite Corson compact space $$X$$, there exists a set $$Q\subset C_{p}(X)$$ with the following properties: (a) $$Q$$ is uniformly dense in $$C_{p} (X)$$; (b) $$\lvert Q \rvert = \omega(X)$$; (c) the set $$Q(x) =\{f(x):\,\,f\in Q\}$$ is countable for any $$x\in X$$. \textbf{4.7 Theorem}. If $$K$$ is an infinite Corson compact space, then there exists a set $$D\subset K$$ such that $$\lvert D\rvert= \omega(K)$$ and $$D$$ has a point-countable open expansion. \textbf{4.9 Theorem}. If $$K$$ is a Corson compact space, then $$C_{p,n}(K)$$ has a uniformly dense subspace $$Y$$ of countable pseudocharacter for any $$n\in\mathbb{N}$$. Reviewer: Ruzinazar Beshimov (Tashkent) Infinite powers and Cohen reals. https://zbmath.org/1459.54008 2021-05-28T16:06:00+00:00 "Medini, Andrea" https://zbmath.org/authors/?q=ai:medini.andrea "van Mill, Jan" https://zbmath.org/authors/?q=ai:van-mill.jan "Zdomskyy, Lyubomyr" https://zbmath.org/authors/?q=ai:zdomskyy.lyubomyr Summary: We give a consistent example of a zero-dimensional separable metrizable space $$Z$$ such that every homeomorphism of $$Z^\omega$$ acts like a permutation of the coordinates almost everywhere. Furthermore, this permutation varies continuously. This shows that a result of \textit{A. Dow} and \textit{E. Pearl} [Proc. Am. Math. Soc. 125, No. 8, 2503--2510 (1997; Zbl 0963.54002)] is sharp, and gives some insight into an open problem of \textit{T. Terada} [Yokohama Math. J. 40, No. 2, 87--93 (1993; Zbl 0819.54006)]. Our example $$Z$$ is simply the set of $$\omega_1$$ Cohen reals, viewed as a subspace of $$2^\omega$$. Almost-normality of Isbell-Mrówka spaces. https://zbmath.org/1459.54015 2021-05-28T16:06:00+00:00 "de Oliveira Rodrigues, Vinicius" https://zbmath.org/authors/?q=ai:rodrigues.vinicius-de-oliveira "dos Santos Ronchim, Victor" https://zbmath.org/authors/?q=ai:dos-santos-ronchim.victor After constructing a Tychonoff, extremally disconnected, pseudocompact space that is not countably compact, the authors turn to properties of the usual Isbell-Mrówka space, i.e. for an almost disjoint family $$\mathcal A$$ over a countable set $$N$$ the space $$\Psi(\mathcal A)$$. A number of conditions equivalent to almost normality of $$\Psi(\mathcal A)$$, such as for each regular closed set $$F$$ there is a clopen set $$C$$ such that $$F\subset C$$ and $$\mathcal A\setminus F\subset\Psi(\mathcal A)\setminus C$$, appear. Conditions equivalent to semi-normality of $$\Psi(\mathcal A)$$ are also presented. Forcing is used to produce an example consistent with \textsf{CH} of a family $$\mathcal A$$ such that $$\Psi(\mathcal A)$$ is almost normal but not normal. For $$\mathfrak p$$ the pseudointersection number, if $$|\mathcal A|<\mathfrak p$$ then $$\Psi(\mathcal A)$$ is semi-normal and is normal if and only if it is almost normal; but if $$|\mathcal A|=\mathfrak c$$ then $$\Psi(\mathcal A)$$ is not semi-normal. Reviewer: David B. Gauld (Auckland)