In the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
field of
general topology
In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...
, a Dowker space is a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
that is
T4 but not
countably paracompact. They are named after
Clifford Hugh Dowker.
The non-trivial task of providing an example of a Dowker space (and therefore also proving their existence as mathematical objects) helped mathematicians better understand the nature and variety of
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s.
Equivalences
Dowker showed, in 1951, the following:
If ''X'' is a normal
T1 space (that is, a
T4 space), then the following are equivalent:
* ''X'' is a Dowker space
* The product of ''X'' with the unit interval is not normal.
* ''X'' is not
countably metacompact
In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological space, there is a refinement that is again an op ...
.
Dowker conjectured that there were no Dowker spaces, and the conjecture was not resolved until
Mary Ellen Rudin
Mary Ellen Rudin (December 7, 1924 – March 18, 2013) was an American mathematician known for her work in set-theoretic topology. In 2013, Elsevier established the Mary Ellen Rudin Young Researcher Award, which is awarded annually to a young res ...
constructed one in 1971. Rudin's counterexample is a very large space (of
cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
).
Zoltán Balogh gave the first
ZFC construction of a small (cardinality
continuum) example, which was more
well-behaved
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved or n ...
than Rudin's. Using
PCF theory PCF theory is the name of a mathematical theory, introduced by Saharon , that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more a ...
, M. Kojman and
S. Shelah
Saharon Shelah (; , ; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.
Biography
Shelah was born in Jerusalem on July 3, 1945. He is the ...
constructed a subspace of Rudin's Dowker space of cardinality
that is also Dowker.
References
{{Reflist
Properties of topological spaces
Separation axioms