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McShane-Whitney extensions in constructive analysis
Within Bishop-style constructive mathematics we study the classical
McShane-Whitney theorem on the extendability of real-valued Lipschitz functions
defined on a subset of a metric space. Using a formulation similar to the
formulation of McShane-Whitney theorem, we show that the Lipschitz real-valued
functions on a totally bounded space are uniformly dense in the set of
uniformly continuous functions. Through the introduced notion of a
McShane-Whitney pair we describe the constructive content of the original
McShane-Whitney extension and examine how the properties of a Lipschitz
function defined on the subspace of the pair extend to its McShane-Whitney
extensions on the space of the pair. Similar McShane-Whitney pairs and
extensions are established for H\"{o}lder functions and -continuous
functions, where is a modulus of continuity. A Lipschitz version of a
fundamental corollary of the Hahn-Banach theorem, and the approximate
McShane-Whitney theorem are shown
The principle of pointfree continuity
In the setting of constructive pointfree topology, we introduce a notion of
continuous operation between pointfree topologies and the corresponding
principle of pointfree continuity. An operation between points of pointfree
topologies is continuous if it is induced by a relation between the bases of
the topologies; this gives a rigorous condition for Brouwer's continuity
principle to hold. The principle of pointfree continuity for pointfree
topologies and says that any relation which induces
a continuous operation between points is a morphism from to
. The principle holds under the assumption of bi-spatiality of
. When is the formal Baire space or the formal unit
interval and is the formal topology of natural numbers, the
principle is equivalent to spatiality of the formal Baire space and formal unit
interval, respectively. Some of the well-known connections between spatiality,
bar induction, and compactness of the unit interval are recast in terms of our
principle of continuity.
We adopt the Minimalist Foundation as our constructive foundation, and
positive topology as the notion of pointfree topology. This allows us to
distinguish ideal objects from constructive ones, and in particular, to
interpret choice sequences as points of the formal Baire space
The computational content of Nonstandard Analysis
Kohlenbach's proof mining program deals with the extraction of effective
information from typically ineffective proofs. Proof mining has its roots in
Kreisel's pioneering work on the so-called unwinding of proofs. The proof
mining of classical mathematics is rather restricted in scope due to the
existence of sentences without computational content which are provable from
the law of excluded middle and which involve only two quantifier alternations.
By contrast, we show that the proof mining of classical Nonstandard Analysis
has a very large scope. In particular, we will observe that this scope includes
any theorem of pure Nonstandard Analysis, where `pure' means that only
nonstandard definitions (and not the epsilon-delta kind) are used. In this
note, we survey results in analysis, computability theory, and Reverse
Mathematics.Comment: In Proceedings CL&C 2016, arXiv:1606.0582
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
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