The Morita conjectures in

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 ...

are certain problems about

normal space Normal(s) or The Normal(s) may refer to: Film and television * Normal (2003 film), ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * Normal (2007 film), ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keit ...

s, now solved in the affirmative. The

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

s, 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 Univer ...

in 1976, asked # If

X \times Y

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

X \times Y

is normal for every normal P-space ''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, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...

? # If

X \times Y

is normal for every normal countably

paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...

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 res ...

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. The axiom is usually written as ''V'' = ''L''. The axiom, first investigated by Kurt Gödel, is inconsistent with the pr ...

''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 {{topology-stub