2025/09/27 by Cortez, María Isabel, Hauser, Till · 1 citation
#37B05 #54B10 (Secondary) #54C10 (Primary) 54B20 #Dynamical Systems (math.DS) #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2509.23015
A continuous surjection π:X→ Y between compact Hausdorff spaces induces continuous surjections M(π)\colon M(X)\toM(Y) and H(π): H(X)\toH(Y) between the spaces of regular Borel probability measures, and the spaces of closed subsets, respetively. It is well known that H(π) is irreducible if and only if π is irreducible. We show that M(π) is irreducible if and only if π is irreducible. Furthermore, we show that whenever π is almost one-to-one then M(π) and H(π) are almost one-to-one. In particular, we observe that continuous surjections between compact metric spaces are almost one-to-one if and only if H(π) is almost one-to-one and a similar statement about M(π). Finally, we give alternative proofs for some results in 'Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps' by Xiongping Dai and Yuxuan Xie regarding semi-open maps.