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Disasters in topology without the axiom of choice

✍ Scribed by Kyriakos Keremedis

Book ID
105842633
Publisher
Springer
Year
2001
Tongue
English
Weight
82 KB
Volume
40
Category
Article
ISSN
0933-5846

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## Abstract We show that the Axiom of Choice is equivalent to each of the following statements: (i) A product of closures of subsets of topological spaces is equal to the closure of their product (in the product topology); (ii) A product of complete uniform spaces is complete.

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## Abstract It is known that – assuming the axiom of choice – for subsets __A__ of ℝ the following hold: (a) __A__ is compact iff it is sequentially compact, (b) __A__ is complete iff it is closed in ℝ, (c) ℝ is a sequential space. We will show that these assertions are not provable in the absence