On coabsoluteness of a space and its exponent (Q1096914)

Topological spaces are called co-absolute if the absolute of these spaces are homeomorphic. In the paper under review the interesting question is investigated as to whether a Tychonoff space X is co-absolute with exp X, where exp X stands for the space of all non-empty compact subsets of X endowed with the Vietoris topology. The author proved that if \(X=\beta N\), then exp X is co-absolute with X since both X and exp X contain countable dense sets of isolated points, but it is not the case for \(X=\beta N-N\) because exp X contains a dense subset of the first category. The main theorem of this paper says that if X is a locally compact rim-compact space with a countable \(\pi\)-base, then exp X is co- absolute with X whenever X is not the topological union of a compact space and a discrete one. Some questions are also stated. One of them was recently answered by L. B. Shapiro who has proved that if X is a dyadic space, then exp X is co-absolute with X.