Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Bolzano-Weierstrass Theorem is the Jump of Weak K\"onig's Lemma   We classify the computational content of the Bolzano-Weierstrass Theorem and
variants thereof in the Weihrauch lattice. For this purpose we first introduce
the concept of a derivative or jump in this lattice and we show that it has
some properties similar to the Turing jump. Using this concept we prove that
the derivative of closed choice of a computable metric space is the cluster
point problem of that space. By specialization to sequences with a relatively
compact range we obtain a characterization of the Bolzano-Weierstrass Theorem
as the derivative of compact choice. In particular, this shows that the
Bolzano-Weierstrass Theorem on real numbers is the jump of Weak K\"onig's
Lemma. Likewise, the Bolzano-Weierstrass Theorem on the binary space is the
jump of the lesser limited principle of omniscience LLPO and the
Bolzano-Weierstrass Theorem on natural numbers can be characterized as the jump
of the idempotent closure of LLPO. We also introduce the compositional product
of two Weihrauch degrees f and g as the supremum of the composition of any two
functions below f and g, respectively. We can express the main result such that
the Bolzano-Weierstrass Theorem is the compositional product of Weak K\"onig's
Lemma and the Monotone Convergence Theorem. We also study the class of weakly
limit computable functions, which are functions that can be obtained by
composition of weakly computable functions with limit computable functions. We
prove that the Bolzano-Weierstrass Theorem on real numbers is complete for this
class. Likewise, the unique cluster point problem on real numbers is complete
for the class of functions that are limit computable with finitely many mind
changes. We also prove that the Bolzano-Weierstrass Theorem on real numbers
and, more generally, the unbounded cluster point problem on real numbers is
uniformly low limit computable. Finally, we also discuss separation techniques.

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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: quant-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
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The categories of this paper are: mtrl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc