Vol. 84, No. 1, 1979

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On the relativization of chain topologies

Marcel Erné

Vol. 84 (1979), No. 1, 43–52

The intrinsic topology J of a chain (X,) induces on any subchain Y X the relative topology J|Y . On the other hand, any such subchain Y is endowed with its own intrinsic topology J|Y . We establish several necessary and sufficient conditions under which both topologies coincide, by suitably weakening the properties of convexity (Lemma 2), order-density (Theorem 3) and subcompleteness (Theorem 4), respectively. Another necessary and sufficient condition for the equation J|Y = J|Y , formulated in terms of cuts, is given in Theorem 2. Besides other related results, we find a purely order-theoretical characterization of those subchains which are compact (Lemma 1) or connected (Corollary 2), respectively, in the intrinsic topology of the entire chain. As a simple consequence of Theorem 4, we obtain the well-known result that the intrinsic topology of a chain can be obtained by relativization from the intrinsic topology of the normal completion (Corollary 9). We conclude with several applications to the Euclidean topology on R.

Mathematical Subject Classification 2000
Primary: 06B30
Secondary: 54B05
Received: 18 October 1978
Published: 1 September 1979
Marcel Erné