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Register a new userAn internal topological characterization of the subspaces of Eberlein compacta and related compacta -- II
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Authors
Georgi D. Dimov
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We generalize a theorem of E. Michael and M. E. Rudin and a theorem of D. Preiss and P. Simon; we give, as well, some partial answers to a recent question of A. V. Arhangelskiv{i}.
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If $g$ is a map from a space $X$ into $mathbb R^m$ and $z otin g(X)$, let $P_{2,1,m}(g,z)$ be the set of all lines $Pi^1subsetmathbb R^m$ containing $z$ such that $|g^{-1}(Pi^1)|geq 2$. We prove that for any $n$-dimensional metric compactum $X$ the functions $gcolon Xtomathbb R^m$, where $mgeq 2n+1$, with $dim P_{2,1,m}(g,z)leq 0$ for all $z otin g(X)$ form a dense $G_delta$-subset of the function space $C(X,mathbb R^m)$. A parametric version of the above theorem is also provided.
We construct an example of a Peano continuum $X$ such that: (i) $X$ is a one-point compactification of a polyhedron; (ii) $X$ is weakly homotopy equivalent to a point (i.e. $pi_n(X)$ is trivial for all $n geq 0$); (iii) $X$ is noncontractible; and (iv) $X$ is homologically and cohomologically locally connected (i.e. $X$ is a $HLC$ and $clc$ space). We also prove that all classical homology groups (singular, v{C}ech, and Borel-Moore), all classical cohomology groups (singular and v{C}ech), and all finite-dimensional Hawaiian groups of $X$ are trivial.
We prove that each metrizable space (of cardinality less or equal to continuum) has a (first countable) uniform Eberlein compactification and each scattered metrizable space has a scattered hereditarily paracompact compactification. Each compact scattered hereditarily paracompact space is uniform Eberlein and belongs to the smallest class of compact spaces, that contain the empty set, the singleton, and is closed under producing the Aleksandrov compactification of the topological sum of a family of compacta from that class.
We study the relations between a generalization of pseudocompactness, named $(kappa, M)$-pseudocompactness, the countably compactness of subspaces of $beta omega$ and the pseudocompactness of their hyperspaces. We show, by assuming the existence of $mathfrak c$-many selective ultrafilters, that there exists a subspace of $beta omega$ that is $(kappa, omega^*)$-pseudocompact for all $kappa<mathfrak c$, but $text{CL}(X)$ isnt pseudocompact. We prove in ZFC that if $omegasubseteq Xsubseteq betaomega$ is such that $X$ is $(mathfrak c, omega^*)$-pseudocompact, then $text{CL}(X)$ is pseudocompact, and we further explore this relation by replacing $mathfrak c$ for some small cardinals. We provide an example of a subspace of $beta omega$ for which all powers below $mathfrak h$ are countably compact whose hyperspace is not pseudocompact, we show that if $omega subseteq X$, the pseudocompactness of $text{CL}(X)$ implies that $X$ is $(kappa, omega^*)$-pseudocompact for all $kappa<mathfrak h$, and provide an example of such an $X$ that is not $(mathfrak b, omega^*)$-pseudocompact.
A function $f:Xto Y$ between topological spaces is called $sigma$-$continuous$ (resp. $barsigma$-$continuous$) if there exists a (closed) cover ${X_n}_{ninomega}$ of $X$ such that for every $ninomega$ the restriction $f{restriction}X_n$ is continuous. By $mathfrak c_sigma$ (resp. $mathfrak c_{barsigma}$) we denote the largest cardinal $kappalemathfrak c$ such that every function $f:Xtomathbb R$ defined on a subset $Xsubsetmathbb R$ of cardinality $|X|<kappa$ is $sigma$-continuous (resp. $barsigma$-continuous). It is clear that $omega_1lemathfrak c_{barsigma}lemathfrak c_sigmalemathfrak c$. We prove that $mathfrak plemathfrak q_0=mathfrak c_{barsigma}=min{mathfrak c_sigma,mathfrak b,mathfrak q}lemathfrak c_sigmalemin{mathrm{non}(mathcal M),mathrm{non}(mathcal N)}$. The equality $mathfrak c_{barsigma}=mathfrak q_0$ resolves a problem from the initial version of the paper.
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