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.

A separable metric space X is called rigid if the identity 1X is the only autohomeomorphism, and homogeneous if, for any points x, y of X, there is an (onto) homeomorphism h: X → X such that h(x) = y.

In this note, we show that this onto-ness of the homeomorphism h could not be removed in the definition of homogeneity, by constructing a continuum X which is rigid and has many embeddings, that is, for any two points x, y, there is an embedding (= into homeomorphism) h: X→X such that h(x) = y.

A space Z is said to have the complete invariance property (CIP) provided that every nonempty closed subset of Z is the fixed point set of some continuous self-mapping of Z. In this paper it is shown that there exists a one-dimensional contractible planar continuum having CIP whose wedge with itself at a specified point is contractible, planar, and does not have CIP.