Strong size properties (Q2843834)

The authors study strong size properties of continua; these are topological properties \(P\) such that, whenever a continuum \(X\) has property \(P\), then so does every strong size level of \(C_n(X)\) for each positive integer \(n\), and they are a natural generalization of Whitney properties. Strong size maps on the \(n\)-fold hyperspace of a continuum were introduced by Hosokawa as a generalization of Whitney maps for the hyperspace of subcontinua of a continuum. Hosokawa proved the existence of such maps and also proved that local connectedness, arcwise connectedness and aposyndesis are strong size properties. In this paper the authors prove that countable aposyndesis, finite-aposyndesis, continuum chainability, acyclicity for \(n \geq 3\), and acyclicity for locally connected continua are strong size properties. They end the paper with a theorem stating that a strong size map defined on a nonempty closed subset of the \(n\)-fold hyperspace can be extended to the complete \(n\)-fold hyperspace.