Pairwise monotonically normal spaces

Marín, Josefa ; Romaguera, Salvador

Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991), p. 567-579 / Harvested from Czech Digital Mathematics Library

Résumé

We introduce and study the notion of pairwise monotonically normal space as a bitopological extension of the monotonically normal spaces of Heath, Lutzer and Zenor. In particular, we characterize those spaces by using a mixed condition of insertion and extension of real-valued functions. This result generalizes, at the same time improves, a well-known theorem of Heath, Lutzer and Zenor. We also obtain some solutions to the quasi-metrization problem in terms of the pairwise monotone normality.

Détail

Publié le : 1991-01-01

MR 1159802 | Zbl 0769.54034

Classification:  54C30,  54D15,  54E20,  54E35,  54E55

Bibliographie

Borges C.R. On stratifiable spaces, Pacific J. Math. 17 (1966), 1-16. (1966) | MR 0188982 | Zbl 0175.19802

Borges C.R. A study of monotonically normal spaces, Proc. Amer. Math. Soc. 38 (1973), 211-214. (1973) | MR 0324644 | Zbl 0257.54018

Borges C.R. Direct sum of stratifiable spaces, Fund. Math. 100 (1978), 97-99. (1978) | MR 0505891

Ceder J.G. Some generalizations of metric spaces, Pacific J. Math. 11 (1961), 105-126. (1961) | MR 0131860 | Zbl 0103.39101

Cooke I.E.; Reilly I.L. On bitopological compactness, J. London Math. Soc. 9 (1975), 518-522. (1975) | MR 0367936 | Zbl 0315.54039

Creede G.D. Concerning semi-stratifiable spaces, Pacific J. Math. 32 (1970), 47-54. (1970) | MR 0254799 | Zbl 0189.23304

Fletcher P.; Hoyle Iii H.B.; Patty C.W. The comparison of topologies, Duke Math. J. 36 (1969), 325-331. (1969) | MR 0242107

Fletcher P.; Lindgren W.F. Quasi-Uniform Spaces, Marcel Dekker, New York, 1982. | MR 0660063 | Zbl 0583.54017

Fox R. On metrizability and quasi-metrizability, preprint.

Gruenhage G. Generalized metric spaces, Handbook of Set-Theoretic Topology (K. Kunen and J.E. Vaughan, Eds.), North-Holland, Amsterdam (1984), 423-501. | MR 0776629 | Zbl 0794.54034

Gutiérrez A. Thesis, Univ. Valencia, 1983.

Gutiérrez A.; Romaguera S. On pairwise stratifiable spaces (in Spanish), Rev. Roumaine Math. P. Appl. 31 (1986), 141-150. (1986) | MR 0839817

Gutiérrez A.; Romaguera S. $C$-binary relations and pairwise normality, Ann. Sci. Univ. ``Al I Cuza'' 32 (1986), 29-33. (1986) | MR 0893680

Heath R.W.; Lutzer D.J.; Zenor P.L. Monotonically normal spaces, Trans. Amer. Math. Soc. 178 (1973), 481-493. (1973) | MR 0372826 | Zbl 0269.54009

Katětov M. Complete normality of Cartesian products, Fund. Math. 36 (1948), 271-274. (1948) | MR 0027501

Katětov M. On real-valued functions in topological spaces, Fund. Math. 38 (1951), 85-91. (1951) | MR 0050264

Kelly J.C. Bitopological spaces, Proc. London Math. Soc. 13 (1963), 71-89. (1963) | MR 0143169 | Zbl 0107.16401

Künzi H.P. Thesis, Univ. Bern, 1981.

Martin J.A.; Marín J.; Romaguera S. On bitopological semi-stratifiability (in Spanish), Proc. XI Jornadas Hispano-Lusas Mat., vol. II (1986/87), 243-251. | MR 0962056

Romaguera S.; Gutiérrez A.; Künzi H.P. Quasi-metrization of pairwise countably compact spaces, Glasnik Mat. 23 (43) (1988), 159-167. (1988) | MR 0976085

Romaguera S.; Salbany S. Quasi-metrization and hereditary normality of compact bitopological spaces, Comment. Math. Univ. Carolinae 31 (1990), 113-122. (1990) | MR 1056178 | Zbl 0698.54026

Singal M.K.; Singal R. Some separation axioms in bitopological spaces, Ann. Soc. Sci. Bruxelles 84 (1970), 207-230. (1970) | MR 0267519

Suzuki J.; Tamano K.; Tanaka Y. $k$-metrizable spaces, stratifiable spaces and metrization, Proc. Amer. Math. Soc. 105 (1989), 500-509. (1989) | MR 0933521

Tong H. Some characterizations of normal and perfectly normal spaces, Duke Math. J. 19 (1952), 289-292. (1952) | MR 0050265 | Zbl 0046.16203

Zenor P.L. Some continuous separation axioms, Fund. Math. 90 (1976), 143-158. (1976) | MR 0394561 | Zbl 0315.54029