##
**A new interval topology for dually directed sets.**
*(English)*
Zbl 0135.22702

Univ. Nac. TucumĂˇn, Rev., Ser. A 14, homenaje a A. Terracini y F. Cernuschi, 325-331 (1962).

Summary: Applying Alexander’s idea [J. W. Alexander, Proc. Natl. Acad. Sci. USA 25, 52–54 (1939; Zbl 0020.16702; JFM 65.1452.02)] of defining a topology for a space in terms of its closed and bounded subsets, to a dually directed set (i.e. a poset in which each pair of elements has an upper bound and a lower bound) the author defines the ‘new interval topology’ as follows: A set which intersects each intersection of finite unions of closed intervals \([a, b]\) in another such set, is closed. Every closed bounded set is such an intersection, and every compact subset is bounded.

The new interval topology is nicer in many respects than the old (see [O. Frink, Trans. Am. Math. Soc. 51, 569–582 (1942; Zbl 0061.39305)]); for example, it gives the usual topology to the plane and permits nontrivial vector lattices to be Hausdorff. The two are equivalent in the presence of a \(0\) and \(1\) but (contrary to Theorem 1) the real line shows that this condition is not necessary. The question of when they coincide is, to my knowledge, still open. (O. Frink, in his review of The paper under review [Math. Rev. 29, No. 3398] asserts their equivalence in all chains.) It is asserted that any conditionally complete lattice is a \(k\)-space in its new interval topology. This, however, is in doubt due to an error in the proof of the purely topological Theorem 2 \((\mathcal K\subseteq \mathcal C^* \) need not hold).

The new interval topology is nicer in many respects than the old (see [O. Frink, Trans. Am. Math. Soc. 51, 569–582 (1942; Zbl 0061.39305)]); for example, it gives the usual topology to the plane and permits nontrivial vector lattices to be Hausdorff. The two are equivalent in the presence of a \(0\) and \(1\) but (contrary to Theorem 1) the real line shows that this condition is not necessary. The question of when they coincide is, to my knowledge, still open. (O. Frink, in his review of The paper under review [Math. Rev. 29, No. 3398] asserts their equivalence in all chains.) It is asserted that any conditionally complete lattice is a \(k\)-space in its new interval topology. This, however, is in doubt due to an error in the proof of the purely topological Theorem 2 \((\mathcal K\subseteq \mathcal C^* \) need not hold).

Reviewer: St. P. Franklin