It is shown that a topological affine Hjelmslev plane is connected or the quasi-component of each point is contained in its neighbour class. If one neighbour class of a point is connected, then they all are, and each is equal to the quasi-component and the component of the point. For topological projective Hjelmslev planes a weaker form of connectedness (∼-connectedness) is defined and it is proved that the plane is ∼-connected or each neighbour class is equal to it ∼-quasi-component. In addition it is shown that the ∼-connectedness of the plane is equivalent to the ∼-connectedness of a line, or other special subsets of the plane, or the connectedness of a line in the associated ordinary plane. Finally it is shown, if the plane is uniform, that ∼-connectedness and connectedness are equivalent and so the plane is either connected, totally disconnected or each neighbour class is equal to the corresponding quasi-component.
J. L. Kelly,Generalized Topology, Van Nostrand (1955).
K. Kuratowski,Topology, Vol. II, New York, Academic Press (1966).
J. W. Lorimer,Coordinate theorems for affine Hjelmslev planes, Ann. di Mat. pura e appl., (IV)105 (1975), pp. 171–180.
J. W. Lorimer,Topological Hjelmslev planes, to appear in Geometriae Dedicata.
H. Lüneburg,Affine Hjelmslev-Ebenen mit transitiver Translationsgrüppe, Math. Z.,79 (1962), pp. 260–288.
H. Salzmann,Über der Zusammenhang in topologischev projectiven Ebenen, Math. Z.61 (1955), pp. 489–494.
O. Wyler,Order and topology in projective planes, Amer. J. Math.,74 (1952), pp. 656–666.
Entrata in Redazione il 19 aprile 1977.
The author gratefully acknowledges the support of the National Research Council of Canada.
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Lorimer, J.W. Connectedness in topological Hjelmslev planes. Annali di Matematica 118, 199–216 (1978) doi:10.1007/BF02415130
- Weak Form
- Special Subset
- Neighbour Class
- Hjelmslev Plane
- Ordinary Plane