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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
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[1] Bell, J.L.; Fremlin, D.H., A geometric form of the axiom of choice, Fund. math., 77, 167-170, (1972) · Zbl 0244.46014
[2] Birkhoff, G., Lattice theory, (1979), American Mathematical Society Providence, RI · Zbl 0126.03801
[3] Cohen, P.J., Set theory and the continuum hypothesis, (1966), New York Benjamin · Zbl 0182.01301
[4] Engelking, R., General topology, (1989), Heldermann Berlin · Zbl 0684.54001
[5] Gartside, P.M., Monotonicity in analytic topology, Ph.D. thesis, (1993), Oxford
[6] Goldblatt, R., On the role of the Baire category theorem and dependent choice in the foundations of logic, J. symbolic logic, 50, 412-422, (1985) · Zbl 0567.03023
[7] Gruenhage, G., Generalized metric spaces, (), 423-501
[8] Haddad, L.; Morillon, M., L’axiome de normalité pour LES espaces totalement ordonnés, J. symbolic logic, 55, 277-283, (1990) · Zbl 0706.03039
[9] Jech, T., The axiom of choice, (1973), North-Holland Amsterdam · Zbl 0259.02051
[10] Jech, T., About the axiom of choice, (), 345-370
[11] Kelly, J.L., The Tychonoff product theorem implies the axiom of choice, Fund. math., 37, 75-76, (1950) · Zbl 0039.28202
[12] Kunen, K., Set theory, an introduction to independence proofs, (1980), North-Holland Amsterdam · Zbl 0443.03021
[13] Läuchli, H., Auswahlaxiom in der algebra, Comment. math. helv., 37, 1-18, (1963) · Zbl 0108.01002
[14] Morillon, M., Extreme choices on complete lexicographic orders, Z. math. logik grundlag. math., 37, 353-355, (1991) · Zbl 0724.03031
[15] Potter, M.D., Sets, an introduction, (1990), Oxford University Press New York · Zbl 0746.04001
[16] Rudin, M.E., A new proof that metric spaces are paracompact, (), 603 · Zbl 0175.49702
[17] Rudin, M.E., Dowker spaces, (), 761-780 · Zbl 0566.54009
[18] Stepr\(a\)ns, J., Trees and continuous mappings into the real line, Topology appl., 12, 181-185, (1981)
[19] Sutherland, W.A., Introduction to metric and topological spaces, (1975), Oxford University Press Oxford · Zbl 0304.54002
[20] Tychonoff, A., Ein fixpuntsatz, Math. ann., 111, 767-776, (1935) · Zbl 0012.30803
[21] van Douwen, E.K., Horrors of topology without AC: a nonnormal orderable space, (), 101-105 · Zbl 0574.03039
[22] Willard, S., General topology, (1970), Addison-Wesley Reading, MA · Zbl 0205.26601
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