Construction of spaces with a -minimal base

Author:
Dennis K. Burke

Journal:
Proc. Amer. Math. Soc. **81** (1981), 329-332

MSC:
Primary 54G99; Secondary 54A25

MathSciNet review:
593483

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Abstract: Using product spaces, a method is given for constructing and recognizing certain spaces with a -minimal base. This technique shows that every topological space can be embedded as a closed subspace of a space with a -minimal base; hence a -minimal base, by itself, does not imply any nontrivial closed hereditary topological property. It is also shown that any space can be expressed as the open perfect image of some space with a -minimal base. Examples are given, illustrating a surprisingly large class of product spaces with a -minimal base.

**[A]**C. E. Aull,*Quasi-developments and 𝛿𝜃-bases*, J. London Math. Soc. (2)**9**(1974/75), 197–204. MR**0388334****[A]**C. E. Aull,*Some properties involving base axioms and metrizability*, TOPO 72—general topology and its applications (Proc. Second Pittsburgh Internat. Conf., Carnegie-Mellon Univ. and Univ. of Pittsburgh, Pittsburgh, Pa., 1972; dedicated to the memory of Johannes H. de Groot), Springer, Berlin, 1974, pp. 41–45. Lecture Notes in Math., Vol. 378. MR**0415575****[BB]**H. R. Bennett and E. S. Berney,*Spaces with 𝜎-minimal bases*, Proceedings of the 1977 Topology Conference (Louisiana State Univ., Baton Rouge, La., 1977), I, 1977, pp. 1–10 (1978). MR**540595****[BL]**H. R. Bennett and D. J. Lutzer,*Ordered spaces with 𝜎-minimal bases*, Proceedings of the 1977 Topology Conference (Louisiana State Univ., Baton Rouge, La., 1977), II, 1977, pp. 371–382 (1978). MR**540616**

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DOI:
https://doi.org/10.1090/S0002-9939-1981-0593483-6

Keywords:
-minimal base,
weight,
product space,
perfect map

Article copyright:
© Copyright 1981
American Mathematical Society