Continuity in separable metrizable and Lindelöf spaces
Authors:
Chris Good and Sina Greenwood
Journal:
Proc. Amer. Math. Soc. 138 (2010), 577591
MSC (2000):
Primary 37B99, 54A10, 54B99, 54C05, 54D20, 54D65, 54H20; Secondary 37XX
Published electronically:
October 14, 2009
MathSciNet review:
2557175
Fulltext PDF
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Abstract: Given a map on a set we examine under what conditions there is a separable metrizable or an hereditarily Lindelöf or a Lindelöf topology on with respect to which is a continuous map. For separable metrizable and hereditarily Lindelöf, it turns out that there is such a topology precisely when the cardinality of is no greater than , the cardinality of the continuum. We go on to prove that there is a Lindelöf topology on with respect to which is continuous if either or for some , where and for any ordinal and limit ordinal .
 1.
D.
Ellis, Orbital topologies, Quart. J. Math., Oxford Ser. (2)
4 (1953), 117–119. MR 0056281
(15,51c)
 2.
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
(91c:54001)
 3.
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
(2007i:54027), http://dx.doi.org/10.1016/j.aim.2005.11.002
 4.
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
0105664 (21 #4402)
 5.
R.
Hodel, Cardinal functions. I, Handbook of settheoretic
topology, NorthHolland, Amsterdam, 1984, pp. 1–61. MR 776620
(86j:54007)
 6.
A.
Iwanik, How restrictive is topological dynamics?, Comment.
Math. Univ. Carolin. 38 (1997), no. 3, 563–569.
MR
1485077 (98k:54078)
 7.
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 (electronic). MR 1906837
(2003f:54052)
 8.
Ludvik
Janos, An application of combinatorial techniques to a topological
problem, Bull. Austral. Math. Soc. 9 (1973),
439–443. MR 0339090
(49 #3853)
 9.
Thomas
Jech, Set theory, Springer Monographs in Mathematics,
SpringerVerlag, Berlin, 2003. The third millennium edition, revised and
expanded. MR
1940513 (2004g:03071)
 10.
Kenneth
Kunen, Set theory, Studies in Logic and the Foundations of
Mathematics, vol. 102, NorthHolland Publishing Co., AmsterdamNew
York, 1980. An introduction to independence proofs. MR 597342
(82f:03001)
 11.
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
(19,1069f)
 1.
 D. Ellis, Orbital topologies, Quart. J. Math. Oxford (2), 4 (1953), 117119. MR 0056281 (15:51c)
 2.
 R. Engelking, General topology, Sigma Series in Pure Mathematics, Vol. 6, Heldermann Verlag, Berlin, 1989. MR 1039321 (91c:54001)
 3.
 C. Good, S. Greenwood, R. W. Knight, D. W. McIntyre, and S. Watson, Characterizing continuous functions on compact spaces, Adv. Math. 206 (2006), 695728. MR 2263719 (2007i:54027)
 4.
 J. de Groot and H. de Vries, Metrization of a set which is mapped into itself, Quart. J. Math. Oxford (2), 9 (1958), 144148. MR 0105664 (21:4402)
 5.
 R. Hodel, Cardinal Functions. I, in K. Kunen and J. E. Vaughan, eds., Handbook of SetTheoretic Topology, NorthHolland, Amsterdam, 1984. MR 776620 (86j:54007)
 6.
 A. Iwanik, How restrictive is topological dynamics? Comment. Math. Univ. Carolin., 38 (1997), 563569. MR 1485077 (98k:54078)
 7.
 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) (electronic), Topology Atlas, North Bay, ON, 2002, 165169. MR 1906837 (2003f:54052)
 8.
 L. Janos, An application of combinatorial techniques to a topological problem, Bull. Austral. Math. Soc., 9 (1973), 439443. MR 0339090 (49:3853)
 9.
 T. Jech, Set Theory, The Third Millennium Edition, SpringerVerlag, Berlin, 2003. MR 1940513 (2004g:03071)
 10.
 K. Kunen, Set Theory, an Introduction to Independence Proofs, NorthHolland, Amsterdam, 1980. MR 597342 (82f:03001)
 11.
 H. de Vries, Compactification of a set which is mapped onto itself, Bull. Acad. Polonaise des Sci. Cl. III, 5 (1957), 943945. MR 0092144 (19:1069f)
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Additional Information
Chris Good
Affiliation:
School of Mathematics, University of Birmingham, Birmingham B15 2TT, United Kingdom
Email:
c.good@bham.ac.uk
Sina Greenwood
Affiliation:
University of Auckland, Private Bag 92019, Auckland, New Zealand
Email:
sina@math.auckland.ac.nz
DOI:
http://dx.doi.org/10.1090/S0002993909101491
PII:
S 00029939(09)101491
Keywords:
Abstract dynamical system,
topological dynamical system,
Lindel\"of,
separable metric,
hereditarily Lindel\"of
Received by editor(s):
October 1, 2008
Published electronically:
October 14, 2009
Communicated by:
Jane M. Hawkins
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
