Connected metrizable subtopologies and partitions into copies of the Cantor set

Irina Druzhinina

Mexico

Universidad Autónoma Metropolitana

Departamento de Matemáticas

Submitted: 2013-11-20

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Accepted: 2013-11-20

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DOI: https://doi.org/10.4995/agt.2006.1919
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Keywords:

Metrizable space, Completely metrizable space, Condensation, Connected metrizable subtopology, Cantor set, Zero-dimensional space, Martin’s Axiom

Supporting agencies:

This research was not funded

Abstract:

We prove under Martin’s Axiom that every separable metrizable space represented as the union of less than 2 ω zero-dimensional compact subsets is zero-dimensional. On the other hand, we show in ZFC that every separable completely metrizable space without isolated points is the union of 2ω pairwise disjoint copies of the Cantor set.

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References:

A.V. Arhangel’skii and V. I. Ponomarev, Fundamentals of General Topology: Problems and Exercises (Translated from the Russian), Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht, 1984. xvi+415 pp.

R. Engelking, General Topology, Helderman Verlag, Berlin 1989.

P. Delaney and W. Just, Two remarks on weaker connected topologies, Comment. Math. Univ. Carolin. 40 no. 2 (1999), 327–329.

I. Druzhinina, Condensations onto connected metrizable spaces, Houston J. Math. 30 no. 3, (2004), 751–766.

K. Kunen, Set Theory, North Holland, 1980.

M.G. Tkachenko, V.V. Tkachuk, V. Uspenskij, and R.G. Wilson, In quest of weaker connected topologies, Comment. Math. Univ. Carolin. 37 no. 4 (1996), 825–841.

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