Connectedness in topological Hjelmslev planes

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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.


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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

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  • Weak Form
  • Special Subset
  • Neighbour Class
  • Hjelmslev Plane
  • Ordinary Plane