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Continuing horrors of topology without choice. (English) Zbl 0822.54001
The authors show that many familiar topological results require some form of the axiom of choice in an essential way. Among the more unnerving results are that, without the axiom of choice, (1) $$\omega_ 1$$ may be paracompact and even Lindelöf; (2) there may be a compact metric space which is neither separable nor second-countable; and (3) there may be a Suslin line whose square is ccc.
Reviewer: K.P.Hart (Delft)

##### MSC:
 54A35 Consistency and independence results in general topology 03E25 Axiom of choice and related propositions 54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 54E35 Metric spaces, metrizability
##### Keywords:
axiom of choice; $$\omega_ 1$$; Suslin line
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