Planar rational compacta

Abstract

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].

Embedding of a planar rational compactum into a planar continuum with the same rim-type

“…It is proved in [1] that Y is a rational containing space for the family of all planar rational compacta. Since X is a planar rational compactum, there exists a homeomorphism h of X onto a subspace of Y .…”

“…Since X is a planar rational compactum, there exists a homeomorphism h of X onto a subspace of Y . Moreover, in [1] the above homeomorphism is constructed in such a manner that condition (1h) is satisfied. Since Y ∩ Int(I 2 ) ∩ V = ∅, condition (2h) holds.…”

“…To do that, we slightly modify the construction of h in [1] applying Lemma 3 of the present paper instead of Lemma 5 of [1].…”