The Morita conjectures in
general topology
In mathematics, general 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 differential topology, geomet ...
are certain problems about
normal space
In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. Th ...
s, now solved in the affirmative. The conjectures, formulated by
Kiiti Morita
was a Japanese mathematician working in algebra and topology.
Morita was born in 1915 in Hamamatsu, Shizuoka Prefecture and graduated from the Tokyo Higher Normal School in 1936. Three years later he was appointed assistant at the Tokyo Univers ...
in 1976, asked
# If
is normal for every normal space ''Y'', is ''X'' a
discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
?
# If
is normal for every normal
P-space
In the mathematical field of topology, there are various notions of a ''P''-space and of a ''p''-space.
Generic use
The expression ''P-space'' might be used generically to denote a topological space satisfying some given and previously introduce ...
''Y'', is ''X''
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
?
# If
is normal for every normal countably
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...
space ''Y'', is ''X'' metrizable and
sigma-locally compact?
The answers were believed to be affirmative. Here a normal P-space ''Y'' is characterised by the property that the product with every metrizable ''X'' is normal; thus the conjecture was that the converse holds.
Keiko Chiba, Teodor C. Przymusiński, and
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 rese ...
proved conjecture (1) and showed that conjectures (2) and (3) cannot be proven false under the standard
ZFC axioms for mathematics (specifically, that the conjectures hold under the
axiom of constructibility
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible universe, constructible. The axiom is usually written as ''V'' = ''L'', where ''V'' and ''L'' denote the von Neumann unive ...
''V=L'').
Fifteen years later,
Zoltán Tibor Balogh Zoltán "Zoli" Tibor Balogh (December 7, 1953 – June 19, 2002) was a Hungarian-born mathematician, specializing in set-theoretic topology. His father, Tibor Balogh, was also a mathematician. His best-known work concerned solutions to problems ...
succeeded in showing that conjectures (2) and (3) are true.
Notes
References
* A.V. Arhangelskii, K.R. Goodearl,
B. Huisgen-Zimmerman, ''Kiiti Morita 1915-1995'', Notices of the AMS, June 199
Topology
Conjectures that have been proved
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