Recent zbMATH articles in MSC 54G20https://zbmath.org/atom/cc/54G202021-05-28T16:06:00+00:00WerkzeugFor every space \(X\), either \(C_{p}C_{p}(X)\) or \(C_{p}C_{p}C_{p}(X)\) is \(\psi\)-separable.https://zbmath.org/1459.540142021-05-28T16:06:00+00:00"Tkachuk, V. V."https://zbmath.org/authors/?q=ai:tkachuk.vladimir-vFor 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/1459.540082021-05-28T16:06:00+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.lyubomyrSummary: 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/1459.540152021-05-28T16:06:00+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.victorAfter 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)