On Closure Operators, the Reals, and Choice

Original Articles

On Closure Operators, the Reals, and Choice

Published in: Quaestiones Mathematicae
Volume 27 , issue 2 , 2004 , pages: 147–151
DOI: 10.2989/16073600409486090
Author(s): Eraldo Giuli , Horst Herrlich
Keywords: AXIOM OF COUNTABLE CHOICE , CLOSURE OPERATOR , SEQUENTIAL SPACE , K -SPACE
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Abstract

The behaviour and the relationships between various closure operators are investigated for subspaces of the reals in the setting of ZF, i. e., Zermelo-Fraenkel set theory without the axiom of choice. Typical results: Equivalent are:

Every subspace of R is a k-space.

The compact closure operator is idempotent for all subspaces of R.

CC (R), the axiom of countable choice for subsets of R, holds.
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