Abstract. We prove that countable aposyndesis, finite-aposyndesis, continuum chainability, acyclicity (for n ≥ 3), and acyclicity for locally connected continua are strong size properties. As a consequence of our results we obtain that arcwise connectedness is a strong size property which is originally proved by Hosokawa.

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Abstract. We continue our study of strong size maps. We show that strong size levels for the n-fold hyperspace of a continuum contain (n − 1)-cells. We give two constructions of strong size maps. We introduce reversible strong size properties. We prove that each of the following properties: being a continuum chainable continuum, being a locally connected continuum, and being a continuum with the property of Kelley, is a reversible strong size property. Following Professors Goodykoontz and Nadler, we define admissible strong size maps and show that the levels of admissible strong size maps for the n-fold hyperspace of a locally connected continuum are homeomorphic to the Hilbert cube. Professor Benjamín Espinoza defined Whitney preserving maps for the hyperspace of subcontinua of a continuum. We define strong size preserving maps and show that this class of maps coincides with the class of homeomorphisms.

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