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: 1. Every subspace of R is a k-space. 2. The compact closure operator is idempotent for all subspaces of R. 3. CC (R), the axiom of countable choice for subsets of R, holds.
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