Connectedness properties of the Higson corona of the half line
The Higson corona of an unbounded proper metric space \(X\) is the remainder \(vX=hX\setminus X\) of the Higson compactification \(hX\) of \(X\). A definition of the Higson compactification can be found, for instance, in [\textit{J. Roe}, Coarse cohomology and index theory on complete Riemannian manifolds. Providence, RI: American Mathematical Society (AMS) (1993; Zbl 0780.58043)] and [\textit{J. Roe}, Lectures on coarse geometry. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1042.53027)]. In this article, the author considers in ZFC the half-line \(\mathbb{H}=\{x\in\mathbb{R}: x\geq 0\}\) equipped with the usual metric \(d\) defined by \(d(x,y)=|x-y|\) for \(x,y\in\mathbb{H}\). Using methods shown in [\textit{K. P. Hart}, in: Recent progress in general topology. Papers from the Prague Toposym 1991, held in Prague, Czechoslovakia, Aug. 19-23, 1991. Amsterdam: North-Holland. 317--352 (1992; Zbl 0847.54028)], the author obtains several interesting results concerning connectedness properties of the Higson corona \(v\mathbb{H}\) in ZFC. In particular, it is proved that \(v\mathbb{H}\) is hereditarily unicoherent, and no non-degenerate subcontinuum of \(v\mathbb{H}\) is hereditarily indecomposable. Furthermore, it is proved that the Continuum Hypothesis implies that the set of all composants of \(v\mathbb{H}\) is of cardinality \(2^{\mathfrak{c}}\). Similarly to the known fact that the Principle of Near Coherence of Filters (denoted by NCF) is equivalent to the statement that \(\beta\mathbb{H}\setminus\mathbb{H}\) has only one composant (see [\textit{A. Blass}, Trans. Am. Math. Soc. 300, 557--581 (1987; Zbl 0647.03043)] and compare with [\textit{J. Mioduszewski}, Proc Conf., Zinnowitz 1974, 257--283 (1978; Zbl 0407.54018)]), it is shown that NCF is equivalent to the assertion that \(v\mathbb{H}\) has exactly one composant. Standard subcontinua of \(v\mathbb{H}\) and layers in them are defined, and it is proved that every layer in a standard subcontinuum of \(v\mathbb{H}\) is an indecomposable continuum. It is shown that if \(K\) is a proper subcontinuum of \(v\mathbb{H}\) and \(O\) is an open subset of \(h\mathbb{H}\) such that \(K\subseteq O\), then there exists a non-degenerate standard subcontinuum \(M\) of \(v\mathbb{H}\) such that \(K\subseteq M\subseteq O\). Finally, it is deduced that every non-empty open subset of \(v\mathbb{H}\) contains a non-degenerate indecomposable continuum. The following question is posed: Are \(v\mathbb{H}\) and \(\beta\mathbb{H}\setminus\mathbb{H}\) homeomorphic?
scientific article; zbMATH DE number 7401527
scientific article