Abstract
This survey’s first object is to introduce the reader to Lindelöf Σ-spaces; since the author would like this introduction to be useful for postgraduate students and non-specialists in the area, most of the basic results are given with complete proofs.
The second object is to make an overview of the recent progress achieved in the study of Lindelöf Σ-spaces. Several popular topics are presented together with open problems, some old and some new. The main idea is to show the areas of major activities displaying what is being done nowadays and trying to outline the trends of their future development. A big percentage of the cited results and problems are new, i.e., published/obtained in the 21-st century. However, some classical old theorems and questions are also discussed the author being convinced that they are worth to be repeated with a new emphasis due to the modern vision of the area.
Resumen
El primer objetivo de este artículo es brindar una introducción a la teoría de los espacios Lindelöf Σ; el autor quisiera que dicha introducción fuera útil tanto para los estudiantes de posgrado como para los que no son especialistas en el área, así que la mayoría de los resultados básicos se presentan con demostraciones completas.
El segundo objetivo es describir, a grandes rasgos, los avances modernos en el estudio de los espacios Lindelöf Σ. Al respecto presentamos algunos temas populares, tanto recientes como ya establecidos desde hace tiempo. La idea principal es mostrar las áreas de mayor actividad, esbozando lo que se está haciendo hoy en día y tratando de visualizar las tendencias para el futuro desarrollo de dichas áreas. Un porcentaje considerable de los resultados citados son nuevos, es decir, obtenidos en el siglo 21. Sin embargo, presentamos también bastantes teoremas clásicos y preguntas abiertas viejas ya que el autor está convencido de que merecen ser mencionados con un nuevo énfasis debido a la visión moderna del área.
Similar content being viewed by others
References
Amir, D. and Lindenstrauss, J., (1968). The structure of weakly compact sets in Banach spaces, Ann. of Math. (2), 88, 1, 35–46.
Alas, O., Tkachuk, V. V. and Wilson, R. G., (2009). A broader context for monotonically monolithic spaces, Acta Math. Hungar., 125, 4, 369–385.
Arhangel’skii, A. V., (1973). On hereditary properties, Gen. Topology and Appl., 8, 39–46.
Arhangel’skii, A. V., (1976). On some topological spaces occurring in functional analysis (in Russian), Uspehi Mat. Nauk, 31, 5, 17–32.
Arhangel’skii, A. V., (1978). Structure and classification of topological spaces and cardinal invariants (in Russian), Uspehi Mat. Nauk, 33, 6, 29–84.
Arhangel’Skii, A. V., (1980). On the relationship between the invariants of topological groups and their subspaces (in Russian), Uspehi Mat. Nauk, 35, 3, 3–22.
Arhangel’skii, A. V., (1981). On d-separable spaces, in Proceedings of the Seminar on General Topology, P. S. Alexandroffed, ed., Mosk. Univ. P. H., 3–8.
Arhangel’skii, A. V., (1983). On relationship between topological properties of X and C p (X), in General Topology and Its Relations to Modern Analysis and Algebra, V, Proc. Fifth Prague Topol. Symp., J. Novak, ed., Heldermannn Verlag, Berlin, 24–36.
Arhangel’skii, A. V., (1984). Continuous mappings, factorization theorems and function spaces (in Russian), Trudy Mosk. Mat. Obsch., 47, 3–21.
Arhangel’skii, A. V., (1987). A survey of C p -theory, Questions Answers Gen. Topology, 5, 1–109.
Arhangel’skii, A. V., (1988). Some results and problems in C p -theory, in General Topology and Its Relations to Modern Analysis and Algebra, VI, Proc. Sixth Prague Topol. Symp., Z. Frolik, ed., Heldermannn Verlag, Berlin, 11–31.
Arhangel’skii, A. V., (1990). Problems in C p -theory, in Open Problems in Topology, North Holland, Amsterdam, 603–615.
Arhangel’skii, A. V., (1992). Topological Function Spaces, Kluwer Acad. Publ., Dordrecht.
Arhangel’skii, A. V., (1992). C p -theory, in Recent Progress in General Topology, M. Hušsek and J. van Mill eds., Elsevier S. P., 1–56.
Arhangel’skii, A. V. and Buzyakova, R. Z., (2002). Addition theorems and D-spaces, Comment. Math. Univ. Carolin., 43, 4, 653–663.
Baturov, D. P., (1987). On subspaces of function spaces (in Russian), Vestnik MGU, Matematika, Mech., 42, 4, 66–69.
Borges, C. R. and Wehrly, A. C. (1991). A study of D-spaces, Topology Proc., 16, 7–15.
Buzyakova, R. Z., (2002). On D-property of strong Σ-spaces, Comment. Math. Univ. Carolin., 43, 3, 493–495.
Buzyakova, R. Z., (2004). Hereditary D-property of function spaces over compacta, Proc. Amer. Math. Soc., 132, 11, 3433–3439.
Buzyakova, R. Z., (2004). In search for Lindelöf C p ’s, Comment. Math. Univ. Carolin., 45, 1, 145–151.
Douwen, Van E. K., (1975). Simultaneous linear extensions of continuous functions, General Topology and its Applications, 5, 297–319.
Douwen, Van E. K. and Lutzer, D. J., (1997). A note on paracompactness in generalized ordered spaces, Proc. Amer. Math. Soc., 125, 1237–1245.
Douwen, Van E. K. and Pfeffer, W. F., (1979). Some properties of the Sorgenfrey line and related spaces, Pacific J. Math., 81, 2, 371–377.
Eisworth, T., (2007). On D-spaces, in Open Problems in Topology II, E. Pearl ed., Elsevier B. V,, Amsterdam, 129–134.
Engelking, R. (1977). General Topology, PWN, Warszawa.
Fabian, M., (1997). Gateaux Differentiability of Convex Functions and Topology, Weak Asplund Spaces, Wiley, New York.
Gruenhage, G., (1984). Generalized metric Spaces, in Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan, eds., North Holland, Amsterdam, 423–501.
Gruenhage, G., (2006). A note on D-spaces, Topology Appl., 153, 2229–2240. DOI: 10.1016/j.topol. 2005.04.012
Gul’ko, S. P., (1979). On the structure of spaces of continuous functions and their complete paracompactness, Russian Math. Surveys., 34, 6, 36–44.
Gillman, L. and Jerison, M., (1960). Rings of Continuous Functions, D. van Nostrand Company Inc., Princeton.
Hodel, R. E., (1975). On a theorem of Arhangel’skii concerning Lindelöf p-spaces, Canad. J. Mat., 27, 459–468.
Hodel, R. E., (1984). Cardinal Functions I, in Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan eds., North Holland, Amsterdam, 1–61.
Juhász, I.; Soukup, L. and Szentmiklóssy, Z., (2007). First countable spaces without point-countable π-bases, Fund. Math., 196, 139–149. DOI: 10.4064/fm196-2-4
Juhász, I. and Szentmiklóssy, Z., (2008). On d-separability of powers and C p (X), Topology Appl., 155, 4, 277–281. DOI: 10.1016/j.topol.2007.09.007
Kubiś, W.; Okunev, O. and Szeptycki, P. J., (2006). On some classes of Lindelöf Σ-spaces, Topology Appl., 153, 2574–2590.
Leiderman, A. G., (1985). On dense metrizable subspaces of Corson compacta (in Russian), Matem. Zametki, 38, 2, 440–449.
Molina Lara, I. and Okunev, O. LΣ(≤ω)-spaces and spaces of continuous functions, Cent. Eur. J. Math., submitted.
Nagami, K., (1969). Σ-spaces, Fund. Math., 65, 2, 169–192.
Neumann, Von J., (1934). Zum Haarschen Maß in topologischen Gruppen, Compos. Math., 1, 1, 106–114.
Okunev, O. G., (1993). On Lindelöf Σ-spaces of continuous functions in the pointwise topology, Topology Appl., 49, 2, 149–166. DOI: 10.1016/0166-8641(93)90041-B
Okunev, O. G., (2007). LΣ(ϰ)-spaces, in Open Problems in Topology II, E. Pearl ed., Elsevier B. V., 47–50.
Okunev, O. G. and Tkachuk, V. V., (2001). Lindelöf Σ-property in C p(X) and p(C p (X)) = ω do not imply countable network weight in X, Acta Math. Hungar., 90, (1–2), 199–132.
Shakhmatov, D. B., (1984). On pseudocompact spaces with a point-countable base, Soviet Math. Doklady, 30, 3, 747–751.
Shakhmatov, D. B., (1986). Precalibers of σ-compact topological groups (in Russian), Matem. Zametki, 39, 6, 859–868.
Shapirovsky, B. E., (1981). Cardinal invariants in compact spaces (in Russian), in Seminar Gen. Topol., P. S. Alexandroff ed., Moscow Univ. P. H., Moscow, 162–187.
Sipacheva, O. V., (1990). The structure of iterated function spaces in the topology of pointwise convergence for Eberlein compacta (in Russian), Matem. Zametki, 47, 3, 91–99.
Sokolov, G. A., (1984). On some classes of compact spaces lying in Σ-products, Comment. Math. Univ. Carolin., 25, 2, 219–231.
Tkachenko, M. G., (1979). On continuous images of dense subspaces of topological products, (in Russian), Uspehi Mat. Nauk, 34, 6, 199–202.
Tkachenko, M. G., (1979). On continuous images of dense subspaces of Σ-products of metrizable compacta, (in Russian), Siberian Math. J., 23, 3, 198–207.
Tkachenko, M. G., (1983). On the Souslin property in free topological groups over compact spaces, (in Russian), Matem. Zametki, 34, 4, 601–607.
Tkachenko, M. G., (1991). Factorization theorems for topological groups and their applications, Topology Appl., 38, 1, 21–37. DOI: 10.1016/0166-8641(91)90038-N
Tkachenko, M. G., (1991). P-approximable compact spaces, Comment. Math. Univ. Carolin., 32, 3, 583–595.
Tkachenko, M. G. and Tkachuk, V. V., (2005). Dyadicity index and metrizability of compact continuous images of function spaces, Topology Appl., 149, 243–257. DOI: 10.1016/j.topol.2004.09.010
Tkachuk, V. V., (1994). A glance at compact spaces which map “nicely” onto the metrizable ones, Topology Proc., 19, 321–334.
Tkachuk, V. V., (2000). Behaviour of the Lindelöf Σ-property in iterated function spaces, Topology Appl., 107, 3, 297–305. DOI: 10.1016/S0166-8641(99)00112-1
Tkachuk, V. V., (2001). Lindelöf Σ-property of C p (X) together with countable spread of X implies X is cosmic, New Zealand J. Math., 30, 93–101.
Tkachuk, V. V., (2005). Point-countable π-bases in first countable and similar spaces, Fund. Math., 186, 1, 55–69. DOI: 10.4064/fm186-1-4
Tkachuk, V. V, (2005). Function spaces and d-separability, Quaest. Math., 28, 409–424.
Tkachuk, V. V., (2007). Condensing function spaces into Σ-products of real lines, Houston J. Math., 33, 1, 209–228.
Tkachuk, V. V., (2009). Monolithic spaces and D-spaces revisited, Topology Appl., 156, 4, 840–846. DOI: 10.1016/j.topol.2008.11.001
Tkachuk, V. V. Some criteria for C p (X) to be an LΣ(≤ω)-space, submitted
Uspenskij, V. V., (1982). A topological group generated by a Lindelöf Σ-space has the Souslin property (in Russian), Doklady AN SSSR, 265, 4, 823–826.
Valdivia, M., (1982). Topics in Locally Convex Spaces, Notas de Matemática, (85), North Holland P. C., Amsterdam.
De La Vega, R. and Kunen, K., (2004). A compact homogeneous S-space, Topology Appl., 136, 123–127. DOI: 10.1016/S0166-8641(03)00215-3
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to professor Manuel Valdivia on the occasion of his 80th birthday
Submitted by José Bonet
Rights and permissions
About this article
Cite this article
Tkachuk, V.V. Lindelöf Σ-spaces: an omnipresent class. RACSAM 104, 221–244 (2010). https://doi.org/10.5052/RACSAM.2010.15
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.5052/RACSAM.2010.15
Keywords
- Compact space
- Eberlein compact
- Corson compact
- Gul’ko space
- Lindelöf Σ-property
- neighbourhood assignments
- Lindelöf property
- D-space
- pseudocompact spaces
- countably compact spaces
- d-separable space
- topological group
- Souslin property