Let X be a continuum. The n-fold hyperspace Cn(X), n < ∞, is the space of all nonempty closed subsets of X with at most n components. A topological property is said to be a (an almost) sequential decreasing strong size property provided that if μ is a strong size map for Cn(X, is a sequence in the interval (t,1) such that lim tj = t ∈ [0,1) (t ∈ (0,1)) and each fiber μ−1(tj) has property , then so does μ−1(t). In this paper we show that the following properties are sequential decreasing strong size properties: being a Kelley continuum, local connectedness, continuum chainability and, unicoherence. Also we prove that indecomposability is an almost sequential decreasing strong size property.
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