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On the spaces of the convex curves in the projective plane Ko, Hwei-Mei

Abstract

Two topologies (Z,L) and (Z,L1) for the family of the non-degenerate convex curves in the projective plane are considered, where (Z,L) is the topology from the Lane's neighborhood system and (Z,L1) is the topology from the parabolic neighborhood system. It is shown that the definition of convexity in the affine plane can be extended to the projective plane so that the Blaschke selection theorem remains true for the projective convex sets. With the help of this theorem, the topological space (Z,L) is compactified by adding Lane's compactifying elements. Furthermore, it is shown that (Z,L) is metrizable but (Z,L1) is not metrizable. The Lane's topology (X,L), as a subspace of (Z,L) for the non-degenerate conics, is both metrizable and separable. A subspace (X,τ) of (Z,L1) is studied which is metrizable but not separable.

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