We study Hausdorff continua in which every set of certain cardinality contains a subset which disconnects the space. We show that such continua are rim-finite. We give characterizations of this class among metric continua. As an application of our methods, we show that continua in which each countably infinite set disconnects are generalized graphs. This extends a result of Nadler for metric continua.

Keywords:

disconnection properties, rim-finite continua, graphs disconnection properties, rim-finite continua, graphs

MSC Classifications:

54D05, 54F20, 54F50 show english descriptions Connected and locally connected spaces (general aspects)
unknown classification 54F20
Spaces of dimension $\leq 1$; curves, dendrites [See also 26A03] 54D05 - Connected and locally connected spaces (general aspects)
54F20 - unknown classification 54F20
54F50 - Spaces of dimension $\leq 1$; curves, dendrites [See also 26A03]

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