850 research outputs found
Pincherle's theorem in Reverse Mathematics and computability theory
We study the logical and computational properties of basic theorems of
uncountable mathematics, in particular Pincherle's theorem, published in 1882.
This theorem states that a locally bounded function is bounded on certain
domains, i.e. one of the first 'local-to-global' principles. It is well-known
that such principles in analysis are intimately connected to (open-cover)
compactness, but we nonetheless exhibit fundamental differences between
compactness and Pincherle's theorem. For instance, the main question of Reverse
Mathematics, namely which set existence axioms are necessary to prove
Pincherle's theorem, does not have an unique or unambiguous answer, in contrast
to compactness. We establish similar differences for the computational
properties of compactness and Pincherle's theorem. We establish the same
differences for other local-to-global principles, even going back to
Weierstrass. We also greatly sharpen the known computational power of
compactness, for the most shared with Pincherle's theorem however. Finally,
countable choice plays an important role in the previous, we therefore study
this axiom together with the intimately related Lindel\"of lemma.Comment: 43 pages, one appendix, to appear in Annals of Pure and Applied Logi
On the relative strength of forms of compactness of metric spaces and their countable productivity in ZF
AbstractWe show in ZF that:(i)A countably compact metric space need not be limit point compact or totally bounded and, a limit point compact metric space need not be totally bounded.(ii)A complete, totally bounded metric space need not be limit point compact or Cantor complete.(iii)A Cantor complete, totally bounded metric space need not be limit point compact.(iv)A second countable, limit point compact metric space need not be totally bounded or Cantor complete.(v)A sequentially compact, selective metric space (the family of all non-empty open subsets of the space has a choice function) is compact.(vi)A countable product of sequentially compact (resp. compete and totally bounded) metric spaces is sequentially compact (resp. compete and totally bounded)
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