Continuous images of H* and its subcontinua (Q1587988)

If \(X\) is the discrete space of integers, \(\mathbb N\), or the half-open interval \([0,1)\) (call this \(\mathbb H\) for half-open or half-line), it is a general problem to succinctly characterize those spaces \(K\) which are remainders. A space \(K\) is a remainder of \(X\) if there is a compact space \(Y\) containing \(X\) densely such that \(K\) is \(Y\smallsetminus X\). In the case of \(\mathbb N\) or \(\mathbb H\), the remainders coincide with the continuous images of the Čech-Stone remainder. For this reason it is also interesting to determine continuous images of various subspaces of \(\beta \mathbb H\smallsetminus \mathbb H\), or \(\mathbb H^*\). This paper makes some interesting contributions by identifying some new remainders of \(\mathbb H\), posing some challenging problems, and comparing the development with that of the remainders of \(\mathbb N\). The paper's main result is to show that every non-trivial subcontinuum of \(\mathbb H^*\) maps onto every continuum of weight at most \(\aleph_1\). The proof nicely adapts a technique of D. P. Bellamy (which was used to prove a similar style of result) together with the fact that \(\mathbb H^*\) itself has been shown to map onto each continuum of weight \(\aleph_1\). The author points out that the technique can be generalized to larger weights so long as \(\mathbb H^*\) maps onto all continua of that weight. The most interesting open problem, possibly not be appearing here for the first time, is whether every separable continuum is a remainder of \(\mathbb H\). It is an interesting contrast to recall that it is quite routine to show that \(\mathbb N^*\) does map onto every separable compactum.