Is every GO-topology a join of two orderable topologies?
HTML articles powered by AMS MathViewer
- by Paul R. Meyer, Victor Neumann Lara and Richard G. Wilson PDF
- Proc. Amer. Math. Soc. 84 (1982), 291-296 Request permission
Abstract:
As a generalization of ${\text {GO}}$-topologies (${\text {GO = generalized ordered}}$), we are interested in those topologies (here called ${\text {JO}}$-topologies) on a set $X$ which can be expressed as a join of orderable topologies (the join being taken in the lattice of all topologies on $X$). If a topology $t$ is the join of $m$ orderable topologies we say $t$ is ${\text {JO}}[m]$. It is not difficult to prove that every ${\text {GO}}$-topology is a ${\text {JO}}$-topology, but the question (raised in [M 71]) as to whether every ${\text {GO}}$-topology is ${\text {JO}}[2]$ seems much more difficult). We show that $X$ is ${\text {JO[2]}}$ if $X$ is a subspace of an orderable space $D$, where $D$ is either metrizable and locally separable, or connected with countable cellularity. (The theorem is actually more general than is stated here.) We give an example to show that for any positive integer $n$ there is a finite join of order topologies which is not ${\text {JO}}[n]$, but these are not ${\text {GO}}$-topologies.References
- Eduard Čech, Topological spaces, Publishing House of the Czechoslovak Academy of Sciences, Prague; Interscience Publishers John Wiley & Sons, London-New York-Sydney, 1966. Revised edition by Zdeněk Frolík and Miroslav Katětov; Scientific editor, Vlastimil Pták; Editor of the English translation, Charles O. Junge. MR 0211373
- Giuseppe De Marco, A characterization of $C(X)$ for $X$ strongly paracompact (or paracompact), Symposia Mathematica, Vol. XXI (Convegno sulle Misure su Gruppi e su Spazi Vettoriali, Convegno sui Gruppi e Anelli Ordinati, INDAM, Rome, 1975), Academic Press, London, 1977, pp. 547–554. MR 0467651
- M. J. Faber, Metrizability in generalized ordered spaces, Mathematical Centre Tracts, No. 53, Mathematisch Centrum, Amsterdam, 1974. MR 0418053
- H. Herrlich, Ordnungsfähigkeit total-diskontinuierlicher Räume, Math. Ann. 159 (1965), 77–80 (German). MR 182944, DOI 10.1007/BF01360281
- D. J. Lutzer, On generalized ordered spaces, Dissertationes Math. (Rozprawy Mat.) 89 (1971), 32. MR 324668
- David J. Lutzer, Book Review: $GO$-spaces and generalizations of metrizability, Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 2, 886–891. MR 1567286, DOI 10.1090/S0273-0979-1980-14841-7
- P. R. Meyer, On total orderings in topology, General topology and its relations to modern analysis and algebra, III (Proc. Third Prague Topological Sympos., 1971) Academia, Prague, 1972, pp. 301–306. MR 0388314
- Paul R. Meyer, The Sorgenfrey topology is a join of orderable topologies, Czechoslovak Math. J. 23(98) (1973), 402–403. MR 319165
- S. Purisch, The orderability and suborderability of metrizable spaces, Trans. Amer. Math. Soc. 226 (1977), 59–76. MR 428296, DOI 10.1090/S0002-9947-1977-0428296-2
- S. Purisch, Scattered compactifications and the orderability of scattered spaces, Topology Appl. 12 (1981), no. 1, 83–88. MR 600466, DOI 10.1016/0166-8641(81)90032-8 R. G. Wilson and P. R. Meyer, Cardinal functions on products of orderable spaces (submitted).
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 84 (1982), 291-296
- MSC: Primary 54F05
- DOI: https://doi.org/10.1090/S0002-9939-1982-0637186-9
- MathSciNet review: 637186