Continuity in separable metrizable and Lindelöf spaces
HTML articles powered by AMS MathViewer
- by Chris Good and Sina Greenwood PDF
- Proc. Amer. Math. Soc. 138 (2010), 577-591 Request permission
Abstract:
Given a map $T:X\to X$ on a set $X$ we examine under what conditions there is a separable metrizable or an hereditarily Lindelöf or a Lindelöf topology on $X$ with respect to which $T$ is a continuous map. For separable metrizable and hereditarily Lindelöf, it turns out that there is such a topology precisely when the cardinality of $X$ is no greater than $\mathfrak {c}$, the cardinality of the continuum. We go on to prove that there is a Lindelöf topology on $X$ with respect to which $T$ is continuous if either $T^{\mathfrak {c}^+}(X)=T^{\mathfrak {c}^++1}(X)\neq \emptyset$ or $T^\alpha (X)=\emptyset$ for some $\alpha <\mathfrak {c}^+$, where $T^{\alpha +1}(X)=T\big (T^\alpha (X)\big )$ and $T^\lambda (X)=\bigcap _{\alpha <\lambda }T^{\alpha }(X)$ for any ordinal $\alpha$ and limit ordinal $\lambda$.References
- D. Ellis, Orbital topologies, Quart. J. Math. Oxford Ser. (2) 4 (1953), 117–119. MR 56281, DOI 10.1093/qmath/4.1.117
- Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
- C. Good, S. Greenwood, R. W. Knight, D. W. McIntyre, and S. Watson, Characterizing continuous functions on compact spaces, Adv. Math. 206 (2006), no. 2, 695–728. MR 2263719, DOI 10.1016/j.aim.2005.11.002
- J. de Groot and H. de Vries, Metrization of a set which is mapped into itself, Quart. J. Math. Oxford Ser. (2) 9 (1958), 144–148. MR 105664, DOI 10.1093/qmath/9.1.144
- R. Hodel, Cardinal functions. I, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1–61. MR 776620
- A. Iwanik, How restrictive is topological dynamics?, Comment. Math. Univ. Carolin. 38 (1997), no. 3, 563–569. MR 1485077
- A. Iwanik, L. Janos, and F. A. Smith, Compactification of a map which is mapped to itself, Proceedings of the Ninth Prague Topological Symposium (2001), Topol. Atlas, North Bay, ON, 2002, pp. 165–169. MR 1906837
- Ludvik Janos, An application of combinatorial techniques to a topological problem, Bull. Austral. Math. Soc. 9 (1973), 439–443. MR 339090, DOI 10.1017/S0004972700043446
- Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
- Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
- H. de Vries, Compactification of a set which is mapped onto itself, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 943–945, LXXIX (English, with Russian summary). MR 0092144
Additional Information
- Chris Good
- Affiliation: School of Mathematics, University of Birmingham, Birmingham B15 2TT, United Kingdom
- MR Author ID: 336197
- ORCID: 0000-0001-8646-1462
- Email: c.good@bham.ac.uk
- Sina Greenwood
- Affiliation: University of Auckland, Private Bag 92019, Auckland, New Zealand
- Email: sina@math.auckland.ac.nz
- Received by editor(s): October 1, 2008
- Published electronically: October 14, 2009
- Communicated by: Jane M. Hawkins
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 577-591
- MSC (2000): Primary 37B99, 54A10, 54B99, 54C05, 54D20, 54D65, 54H20; Secondary 37-XX
- DOI: https://doi.org/10.1090/S0002-9939-09-10149-1
- MathSciNet review: 2557175