Higson compactifications obtained by expanding and contracting the half-open interval (Q5949483)

All spaces are assumed to be Hausdorff and locally compact. Let \((X,d)\) be metric space, let \(C^*(X)\) be the Banach algebra of all bounded real-valued continuous functions on \(X\) with the sup-norm. Let \(C^*_d(X)\) be the (closed) subring of \(C^*(X)\) of functions \(f\) such that for each \(r > 0\) and each \(\varepsilon > 0\), there is in \(X\) a compact set \(K_{r,\varepsilon}\) such that \(\text{diam} f(B_d(x,r)) < \varepsilon\) for each \(x\in X \setminus K\). The Higson compactification \(\overline{X}^d\) of a proper metric space \((X, d)\) (every bounded subset has a compact closure) is the compactification associated with the closed subring \(C^*_d(X)\). The remainder \(\overline{X}^d \setminus X\) is called the Higson corona of \(X\). Typical result (Theorem 2.2): Let \(X\) be the half-open interval. Then the Stone-Čech compactification \(\beta X\) can be approximated by Higson compactifications whose coronas are indecomposable continua.