1. Introduction. In this paper we consider rational subspaces of the plane. A rational space is a space which has a basis of open sets with countable boundaries. In the special case where the boundaries are finite, the space is called rim-finite.G. Nöbeling [8] has proved that the family of all rim-finite spaces does not contain a universal element. The same is true even for the family of planar rim-finite spaces. This fact is included in a wider result (see [1] and [4]) concerning some families of planar rim-scattered spaces.S. Iliadis [3] (see also [7]) proved that there exists a universal rational space. Therefore there exists a rational space which contains topologically all rational compacta.In [6] J. Mayer and E. Tymchatyn constructed a planar continuum of rim-type α+1 which is a containing space for all planar compacta of rim-type ≤ α, where α is a countable ordinal.In this paper we give a simple, direct and visualized example of a planar rational connected and locally connected space which is a containing space for all planar rational compacta. This provides an affirmative answer to Problem 5(2) of [2].
We investigate the problem of existence of universal elements in some families of dendrites with a countable closure of the set of end points. In particular, we prove that for each integer κ 3 and for each ordinal α 1 there exists a universal element in the family of all dendrites X such that ord( X) κ and the α-derivative of the set cl X E( X) contains at most one point.
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