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.
