scholarly journals BROUWER’S FAN THEOREM AND CONVEXITY

2018 ◽  
Vol 83 (04) ◽  
pp. 1363-1375 ◽  
Author(s):  
JOSEF BERGER ◽  
GREGOR SVINDLAND
Keyword(s):  
Continuous Functions ◽  
Unit Interval ◽  
Binary Sequences ◽  
Constructive Mathematics ◽  
Convexity Condition ◽  
Fan Theorem ◽  
Image Position ◽  
Uniformly Continuous ◽  
Bishop’S Constructive Mathematics

AbstractIn the framework of Bishop’s constructive mathematics we introduce co-convexity as a property of subsets B of ${\left\{ {0,1} \right\}^{\rm{*}}}$, the set of finite binary sequences, and prove that co-convex bars are uniform. Moreover, we establish a canonical correspondence between detachable subsets B of ${\left\{ {0,1} \right\}^{\rm{*}}}$ and uniformly continuous functions f defined on the unit interval such that B is a bar if and only if the corresponding function f is positive-valued, B is a uniform bar if and only if f has positive infimum, and B is co-convex if and only if f satisfies a weak convexity condition.

How connected is the intuitionistic continuum?

1997 ◽  
Vol 62 (4) ◽  
pp. 1147-1150 ◽  
Author(s):  
Dirk Van Dalen
Keyword(s):  
Real Function ◽  
Peculiar Feature ◽  
Constructive Mathematics ◽  
Fan Theorem ◽  
First Time ◽  
Uniformly Continuous ◽  
Intuitionistic Mathematics ◽  
Excluded Middle ◽  
Do So ◽  
The Continuum

In the twenties Brouwer established the well-known continuity theorem “every real function is locally uniformly continuous,” [3, 2, 5]. From this theorem one immediately concludes that the continuum is indecomposable (unzerlegbar), i.e., if ℝ = A ∪ B and A ∩ B = ∅ (denoted by ℝ = A + B), then ℝ = A or ℝ = B.Brouwer deduced the indecomposability directly from the fan theorem (cf. the 1927 Berline Lectures, [7, p. 49]).The theorem was published for the first time in [6], it was used to refute the principle of the excluded middle: ¬∀x ∈ ℝ (x ∈ ℚ ∨ ¬x ∈ ℚ).The indecomposability of ℝ is a peculiar feature of constructive universa, it shows that ℝ is much more closely knit in constructive mathematics, than in classically mathematics. The classically comparable fact is the topological connectedness of ℝ. In a way this characterizes the position of ℝ: the only (classically) connected subsets of ℝ are the various kinds of segments. In intuitionistic mathematics the situation is different; the continuum has, as it were, a syrupy nature, one cannot simply take away one point. In the classical continuum one can, thanks to the principle of the excluded third, do so. To put it picturesquely, the classical continuum is the frozen intuitionistic continuum. If one removes one point from the intuitionistic continuum, there still are all those points for which it is unknown whether or not they belong to the remaining part.

Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups

1986 ◽  
Vol 99 (2) ◽  
pp. 273-283 ◽  
Author(s):  
Anthony To-Ming Lau
Keyword(s):  
Continuous Functions ◽  
Supremum Norm ◽  
Hausdorff Topology ◽  
Closed Subalgebra ◽  
Continuous Complex ◽  
Arens Multiplication ◽  
Image Position ◽  
Bounded Continuous Functions ◽  
Complex Valued ◽  
Uniformly Continuous

Let G be a topological semigroup, i.e. G is a semigroup with a Hausdorff topology such that the map (a, b) → a.b from G × G into G is continuous when G × G has the product topology. Let C(G) denote the space of complex-valued bounded continuous functions on G with the supremum norm. Let LUC (G) denote the space of bounded left uniformly continuous complex-valued functions on G i.e. all f ε C(G) such that the map a → laf of G into C(G) is continuous when C(G) has a norm topology, where (laf )(x) = f (ax) (a, x ε G). Then LUC (G) is a closed subalgebra of C(G) invariant under translations. Furthermore, if m ε LUC (G)*, f ε LUC (G), then the functionis also in LUC (G). Hence we may define a productfor n, m ε LUC(G)*. LUC (G)* with this product is a Banach algebra. Furthermore, ʘ is precisely the restriction of the Arens product defined on the second conjugate algebra l∞(G)* = l1(G)** to LUC (G)*. We refer the reader to [1] and [10] for more details.

Two questions from Dana Scott: Intuitionistic topologies and continuous functions

2009 ◽  
Vol 74 (2) ◽  
pp. 689-692
Author(s):  
Charles McCarty
Keyword(s):  
Set Theory ◽  
Continuous Functions ◽  
Unit Interval ◽  
Natural Numbers ◽  
Double Negation ◽  
Closed Set ◽  
Closed Sets ◽  
Image Position ◽  
Intuitionistic Set Theory ◽  
Formal Theories

Since intuitionistic sets are not generally stable – their membership relations are not always closed under double negation – the open sets of a topology cannot be recovered from the closed sets of that topology via complementation, at least without further ado. Dana Scott asked, first, whether it is possible intuitionistically for two distinct topologies, given as collections of open sets on the same carrier, to share their closed sets. Second, he asked whether there can be intuitionistic functions that are closed continuous in that the inverse of every closed set is closed without being continuous in the usual, open sense. Here, we prove that, as far as intuitionistic set theory is concerned, there can be infinitely-many distinct topologies on the same carrier sharing a single collection of closed sets. The proof employs Heyting-valued sets, and demonstrates that the intuitionistic set theory IZF [4, 624], as well as the theory IZF plus classical elementary arithmetic, are both consistent with the statement that infinitely many topologies on the set of natural numbers share the same closed sets. Without changing models, we show that these formal theories are also consistent with the statement that there are infinitely many endofunctions on the natural numbers that are closed continuous but not open continuous with respect to a single topology.For each prime k ∈ ω, let Ak be this ω-sequence of sets open in the standard topology on the closed unit interval: for each n ∈ ω,

scholarly journals Heine-Borel does not imply the Fan Theorem

1984 ◽  
Vol 49 (2) ◽  
pp. 514-519 ◽  
Author(s):  
Ieke Moerdijk
Keyword(s):  
Function Space ◽  
Real Line ◽  
General Framework ◽  
Geometric Theory ◽  
Continuous Functions ◽  
Unit Interval ◽  
Baire Space ◽  
Locally Compact ◽  
Cantor Space ◽  
Fan Theorem

This paper deals with locales and their spaces of points in intuitionistic analysis or, if you like, in (Grothendieck) toposes. One of the important aspects of the problem whether a certain locale has enough points is that it is directly related to the (constructive) completeness of a geometric theory. A useful exposition of this relationship may be found in [1], and we will assume that the reader is familiar with the general framework described in that paper.We will consider four formal spaces, or locales, namely formal Cantor space C, formal Baire space B, the formal real line R, and the formal function space RR being the exponential in the category of locales (cf. [3]). The corresponding spaces of points will be denoted by pt(C), pt(B), pt(R) and pt(RR). Classically, these locales all have enough points, of course, but constructively or in sheaves this may fail in each case. Let us recall some facts from [1]: the assertion that C has enough points is equivalent to the compactness of the space of points pt(C), and is traditionally known in intuitionistic analysis as the Fan Theorem (FT). Similarly, the assertion that B has enough points is equivalent to the principle of (monotone) Bar Induction (BI). The locale R has enough points iff its space of points pt(R) is locally compact, i.e. the unit interval pt[0, 1] ⊂ pt(R) is compact, which is of course known as the Heine-Borel Theorem (HB). The statement that RR has enough points, i.e. that there are “enough” continuous functions from R to itself, does not have a well-established name. We will refer to it (not very imaginatively, I admit) as the principle (EF) of Enough Functions.

Boundaries for polydisc algebras in infinite dimensions

1979 ◽  
Vol 85 (2) ◽  
pp. 291-303 ◽  
Author(s):  
J. Globevnik
Keyword(s):  
Unit Ball ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Open Unit ◽  
Closed Unit Ball ◽  
Infinite Dimensions ◽  
Open Unit Ball ◽  
Image Position ◽  
Bounded Continuous Functions ◽  
Uniformly Continuous

AbstractLet AB be the algebra of all bounded continuous functions on the closed unit ball B of c0, analytic on the open unit ball, with sup norm, and let AU be the sub-algebra of AB of those functions which are uniformly continuous on B. Call a set S ⊂ B a boundary of AB (AU) iffor every f ∈ AB (f ∈AU, respectively). In the paper we study the boundaries of AB and AU. We give a complete description of the boundaries of AU and present some necessary and some sufficient conditions for a set to be a boundary of AB. We also give some examples of boundaries.

scholarly journals Lattices of uniformly continuous functions

2013 ◽  
Vol 160 (1) ◽  
pp. 50-55 ◽  
Author(s):  
Félix Cabello Sánchez ◽  
Javier Cabello Sánchez
Keyword(s):  
Continuous Functions ◽  
Uniformly Continuous

Periodic solutions to functional differential equations

1985 ◽  
Vol 101 (3-4) ◽  
pp. 253-271 ◽  
Author(s):  
O. A. Arino ◽  
T. A. Burton ◽  
J. R. Haddock
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Ultimate Boundedness ◽  
Space Of Continuous Functions ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential ◽  
Image Position

SynopsisWe consider a system of functional differential equationswhere G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

scholarly journals SEPARATING THE FAN THEOREM AND ITS WEAKENINGS

2014 ◽  
Vol 79 (3) ◽  
pp. 792-813 ◽  
Author(s):  
ROBERT S. LUBARSKY ◽  
HANNES DIENER
Keyword(s):  
Constructive Mathematics ◽  
Kripke Models ◽  
Fan Theorem

AbstractVarieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Some of the implications have been shown to be strict, others strict in a weak context, and yet others not at all, using disparate techniques. Here we present a family of related Kripke models which separates all of the as yet identified fan theorems.

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