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converge 0-regularly

A sequence of sets $\{A_n\}$$ contained in a metric space

with a metric

is said to converge 0-regularly to its limit $A = \mathrm{Lim}\,A_n$$ (see hyperspace ) provided that for each $\varepsilon > 0$$ there is a $\delta>0$$ and there is an index $n_0 \in \mathbb{N}$$ such that if

then for every two points $p, q \in A_n$$ with $d(p,q) < \delta$$ there is a connected set $C_n \subset A_n$$ satisfying conditions $p, q \in C_n$$ and $diam_n < \varepsilon $$ (see [Whyburn 1942, Chapter 9, §3, p. 174]).

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Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30