Let X denote a locally connected continuum such that cyclic elements have metrizable boundary in X. We study the cyclic elements of X by demonstrating that each such continuum gives rise to an upper semicontinuous decomposition G of X into continua such that X/G is the continuous image of an arc and the cyclic elements of X correspond to the cyclic elements of X/G that are Peano continua.
Let X be a connected locally compact metric space. It is known that if X is locally connected, then the space of ends (Freudenthal ends), EX, can be represented as the inverse limit of the set (space) S(X C) of components of X C, i.e., as the inverse limit of the inverse system . In this paper, the above result is significantly improved. It is shown that for a space which is not locally connected, we can replace the space of components by the space of quasicomponents Q(X C) of X C. The following...
We deal with a hyperspace selection problem in the setting of connected spaces. We present two solutions of this problem illustrating the difference between selections for the nonempty closed sets, and those for the at most two-point sets. In the first case, we obtain a characterisation of compact orderable spaces. In the latter case --- that of selections for at most two-point sets, the same selection property is equivalent to the existence of a ternary relation on the space, known as a cyclic...
We study when a topological space has a weaker connected topology. Various sufficient and necessary conditions are given for a space to have a weaker Hausdorff or regular connected topology. It is proved that the property of a space of having a weaker Tychonoff topology is preserved by any of the free topological group functors. Examples are given for non-preservation of this property by “nice” continuous mappings. The requirement that a space have a weaker Tychonoff connected topology is rather...