##
**Set theory in topology.**
*(English)*
Zbl 0796.54001

Hušek, Miroslav (ed.) et al., Recent progress in general topology. Papers from the Prague Toposym 1991, held in Prague, Czechoslovakia, Aug. 19-23, 1991. Amsterdam: North-Holland. 167-197 (1992).

The paper surveys a number of set-theoretic techniques that, in the author’s view, in recent years have played an important role in general topology.

Many of these techniques involve some kind of reflection.

One general method is to use elementary submodels in proofs and constructions, rather than elaborate transfinite inductions. This is illustrated by means of (a version of) M. E. Rudin’s example of a normal non-collectionwise Hausdorff space [Proc. Am. Math. Soc. 91, 155- 158 (1984; Zbl 0554.54006)]; this space is built by considering via elementary submodels all possible countable approximations to it and thus preventing it from being collectionwise Hausdorff.

In iterated forcing constructions one usually arrives at the desired conclusion by assuming its negation, reflecting this to an intermediate stage and showing that it had been dealt with. Again elementary submodels free us from the burden of transfinite induction. By way of example the author discusses Balogh’s result that locally compact normal spaces are collectionwise normal after supercompact many Cohen reals have been added to the universe.

Other interesting material includes application of PFA and OCA and a section on Kurepa families by S. Todorčević.

For the entire collection see [Zbl 0782.00072].

Many of these techniques involve some kind of reflection.

One general method is to use elementary submodels in proofs and constructions, rather than elaborate transfinite inductions. This is illustrated by means of (a version of) M. E. Rudin’s example of a normal non-collectionwise Hausdorff space [Proc. Am. Math. Soc. 91, 155- 158 (1984; Zbl 0554.54006)]; this space is built by considering via elementary submodels all possible countable approximations to it and thus preventing it from being collectionwise Hausdorff.

In iterated forcing constructions one usually arrives at the desired conclusion by assuming its negation, reflecting this to an intermediate stage and showing that it had been dealt with. Again elementary submodels free us from the burden of transfinite induction. By way of example the author discusses Balogh’s result that locally compact normal spaces are collectionwise normal after supercompact many Cohen reals have been added to the universe.

Other interesting material includes application of PFA and OCA and a section on Kurepa families by S. Todorčević.

For the entire collection see [Zbl 0782.00072].

Reviewer: E.Coplakova (Delft)

### MSC:

54-02 | Research exposition (monographs, survey articles) pertaining to general topology |

03C99 | Model theory |

54A35 | Consistency and independence results in general topology |

03E35 | Consistency and independence results |

03E55 | Large cardinals |