Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Categorical concepts and their generalization by means of game theory
  (german: Kategorien-Konzepte und ihre spieltheoretische Verallgemeinerung)   This paper deals with different concepts for characterizing the size of
mathematical objects. A game theoretic investigation and generalization of two
size concepts, which can both be formulated in topological terms, is provided:
the so called "Baire category" and the "$\sigma $-category". This is mainly
done by means of (generalized) Banach-Mazur games using the Axiom of
determinacy (the inconsistency of AC and AD is reflected in the beginning and a
weaker form of AC is chosen for proofs). Analogue versions of Cantor-Bendixson
and Heine-Borel are proofed as well as some definability results.
  Such size concepts are established f.i. in measure and integration theory,
set theory and topology often leading to mathematically precise formulations of
"fuzziness". In measure and integration theory f.i. one defines for a "measure
space" $(\Omega, \underline{A}, \mu)$ - i.e. $\Omega$ is a non empty set,
$\underline{A}$ a $\sigma$-algebra on $\Omega$ and $\mu$ a measure on
$(\Omega,\underline{A})$ - and a property $E \subset \Omega$ of elements of
$\Omega$, that the property $E$ is valid in a set $A\in\underline{A}$
"$\mu$-almost everywhere", if $E$ is true in $A$ up to a "$\mu$-null set" -
i.e. $\exists N\in\underline{A}{ } (\mu(N)=0 \wedge A\cap N^\complement \subset
E)$. This is crucial for the formulation of uniqueness statements in measure
and integration theory as its theorems usually only apply up to "small"
($\mu$-null) sets.
  Key words: Axiom of choice AC, Axiom of determinacy AD, Baire category
theorem, Baire property, Baire space, Banach-Mazur games, Borel hierachy,
Cantor-Bendixson, definability, Gale-Stewart, Heine-Borel, Lusin hierarchy,
meager sets, perfect set property, polish spaces, projective hierarchy, sigma
bounded sets, sigma compact sets, superperfect sets, topological games, winning
strategy

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The categories of this paper are: math-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-th
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The categories of this paper are: econ
1
The categories of this paper are: cs
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ex
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The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys