Cardinal Functions, Revised
Ofelia Teresa Alas
University of Sao Paulo,
Institute of Mathematics and Statistics,
Caixa Postal 66281,
05315-970, Sao Paulo SP,
Brazil
alas@ime.usp.br
The idea is to associate to each non-empty topological space X some infinite
cardinal numbers (which are invariant under homeomorphisms) and to establish
relations among them, depending on topological properties satisfied by X.
There are many cardinal functions, we shall recall a few definitions and results.
Definitions
Let X be a non-empty topological space.
The character at a point p in X is the smallest cardinal of a local basis at p
(it may be finite) and the character of X is the smallest infinite
cardinal number k such that the
character at each point of X is at most k.
The weight of X is the smallest infinite cardinal number
k such that X has an open basis of
cardinality at most k.
The p-character at a point p in X is the
smallest cardinal number k such that
there is a non-empty collection C of non-empty open sets
of cardinality at most k such that
if U is an open neighborhood of p, there is a V in C such that
V is a subset of U.
Results
Let X be a non-empty Hausdorff space and k
be an infinite cardinal number.
- If X is compact and the character at each point of X is at least
k,
then the cardinality of X is at least
2k
(Cech-Pospišil-1937)
- If the character of X is at most k
and each open cover of X has a subcover of cardinality at most
k,
than the cardinality of X is at most 2k.
(Arhangel'skii-1969)
- If X is an infinite completely metrizable space whose weight is
k,
then the cardinality of X is equal either to
k
or
k\aleph_0.
(A.H. Stone-1959)
- If X is compact and there is no continuous function from X onto
[0,1]k,
then the set of points of X whose
p-character is k
is dense in X. (Sapirovskii-1975)
Since 1965 many interesting results have been proved by applying
combinatorial principles and set-theoretic topology methods.
See, for instance,
the survey by I. Juhász, Cardinal functions II in the
Handbook of Set-Theoretic Topology, K. Kunen and J.E. Vaughan, Eds.,
(North-Holland, Amsterdam 1984), 63-109.
Another (more recent) line of research deals with reflection.
One of the questions is the following: given an infinite cardinal number
k, a non-empty topological space X, a
cardinal function f
and a non-empty collection C of subspaces of X,
if f(Y) is at most k,
for all Y in C, is it true that
f(X) is at most k?
Let us recall some results on reflection
- A compact Hausdorff space is metrizable if and only if
all of its subspaces of size at most \aleph_1 are metrizable
(A. Dow - 1988)
- A Hausdorff space X has weight at least k
if and only if
there is a subset Y of X, with cardinality not bigger than k,
and weight of Y is at least k
(Hajnal & Juhász-1980)
- A compact Hausdorff space is first countable if and only if all of its
subspaces of size at most \aleph_1 are first countable.
(Hodel & Vaughn - 2000).
- If X is a Lindelöf Hausdorff space in which every subspace of size
at most \aleph_1 is first countable,
then X has size at most 2\aleph_0 and each point is
Gd.
(Juhász-2000?)
- Recently F. Tall and P. Koszmider anounced a consistent example of a Hausdorff
Lindelöf space with no Lindelöf subspace of size \aleph_1.
Other References (Surveys)
- I. Juhász, Cardinal functions. Ten years later,
Math. Centrum Tract 23, Amsterdam 1980.
- This book is a classical reference on the subject.
- R. Hodel, Cardinal functions I,
in: Handbook of Set-Theoretic
Topology, K. Kunen and J.E. Vaughan, Eds.,
(North-Holland, Amsterdam (1984), 1-61.
- Very good survey, where definitions and inequalities are easy to find.
- A.V. Arhangel'skii, A survey of Cp-theory, Q & A in General
Topology 5 (1987), 1-109.
- The author deals mainly with cardinal functions on the space of continuous functions.
- R.E. Hodel & J.E. Vaughn,
Reflection theorems for cardinal functions, Top. Appl. 100
(2000), 47-66.
- Extensive survey on reflections results.
Received: December 18, 1995
Revised: December 27, 1995, December 15, 2000.
Copyright 2000 © Topology Atlas.