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**On the axioms preserved by modifications of topologies without axioms.**
*(English)*
Zbl 0681.54003

[This is a joint review with the preceding paper.]

Over the years various axioms for topological closure operators have been studied. Following K. Koutsky and M. Sekanina, the present author undertakes the most general possible approach by calling any self mapping of exp P a topology without axioms on P. In this setting every theorem can be stated in terms of precisely which axioms are needed for its proof. As such a general mathematical structure, these operators arise in diverse areas. (An example from logic: If P is the set of all formulas of some fixed formal logic, then the consequence operation defines a closure operation on exp P.) If f is a topological property, then a topology possessing f is called an f-topology. The coarsest (finest) of all f- topologies which are finer (coarser) than a given topology u is called the lower (upper) f-modification of u. The present papers are concerned primarily with questions about f-modifications, where f is one of the standard axioms for a closure operator.

Over the years various axioms for topological closure operators have been studied. Following K. Koutsky and M. Sekanina, the present author undertakes the most general possible approach by calling any self mapping of exp P a topology without axioms on P. In this setting every theorem can be stated in terms of precisely which axioms are needed for its proof. As such a general mathematical structure, these operators arise in diverse areas. (An example from logic: If P is the set of all formulas of some fixed formal logic, then the consequence operation defines a closure operation on exp P.) If f is a topological property, then a topology possessing f is called an f-topology. The coarsest (finest) of all f- topologies which are finer (coarser) than a given topology u is called the lower (upper) f-modification of u. The present papers are concerned primarily with questions about f-modifications, where f is one of the standard axioms for a closure operator.

Reviewer: P.R.Meyer

### MSC:

54A10 | Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) |

54A05 | Topological spaces and generalizations (closure spaces, etc.) |