## Located Sets And Reverse Mathematics (1999)

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Venue: | Journal of Symbolic Logic |

Citations: | 14 - 5 self |

### BibTeX

@ARTICLE{Giusto99locatedsets,

author = {Mariagnese Giusto and Stephen G. Simpson},

title = {Located Sets And Reverse Mathematics},

journal = {Journal of Symbolic Logic},

year = {1999},

volume = {65},

pages = {1451--1480}

}

### Years of Citing Articles

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### Abstract

Let X be a compact metric space. A closed set K is located if the distance function d(x, K) exists as a continuous realvalued function on X ; weakly located if the predicate d(x, K) > r is # 1 allowing parameters. The purpose of this paper is to explore the concepts of located and weakly located subsets of a compact separable metric space in the context of subsystems of second order arithmetic such as RCA 0 , WKL 0 and ACA 0 . We also give some applications of these concepts by discussing some versions of the Tietze extension theorem. In particular we prove an RCA 0 version of this result for weakly located closed sets.

### Citations

500 |
Recursively enumerable sets and degrees
- Soare
- 1978
(Show Context)
Citation Context ...ction 1. On the other hand, to prove that (5) =# (1) is not so easy. To discuss more in detail this problem we recall some definitions and notations of recursion theory (for more details see [12] and =-=[18]-=-). Let p # Q[x] be a polynomial with rational coe#cients; the code for p is given by #(p). The code for f # C[0, 1] # REC is given by a sequence #p n : n # N# of polynomials p n # Q[x] such that #f - ... |

206 |
Classical Descriptive Set Theory, Graduate Texts
- Kechris
- 1995
(Show Context)
Citation Context ... n # N (1) I = I 0 (n) #s# I k (n) (2) #j # k (m # I j (n) =# # j (n, m)). 3. K(X) and Located Sets Let X be a compact complete separable metric space. In the literature of general topology (see e.g. =-=[11]-=- and [5]), the space of nonempty compact subsets of X is known as K(X). It is usually equipped with the Vietoris topology, generated by sets of the form {K # K(X) : K # U} and {K # K(X) : K # U #= #} ... |

58 |
Constructive analysis, Grundlehren der
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ...of a compact complete separable metric space. This concept arises naturally in the context of metric spaces. Even if with a di#erent aim, it plays a fundamental role in the work of Bishop and Bridges =-=[1]-=-. Bishop and Bridges proved a constructive version of the well known Tietze extension theorem for located closed sets in a compact space and uniformly continuous functions with modulus of uniform cont... |

14 |
Which set existence axioms are needed to prove the separable Hahn-Banach theorem
- Brown, Simpson
- 1986
(Show Context)
Citation Context ...ear from the context the object which we are referring to. Notice that for each n # N the n-net is a covering of the space X. Continuous functions are coded in second order arithmetic as follows (see =-=[4, 17]-=-). Definition 2.7 (RCA 0 ). Let b A and b B be two complete separable metric spaces. A (code for a) continuous function from b A to b B is a set # # NAQ + B Q + such that, if we denote by (a, r)#(b, s... |

13 |
Theory of Recursive Functions and Effective
- Rogers
(Show Context)
Citation Context ...rem in section 1. On the other hand, to prove that (5) =⇒ (1) is not so easy. To discuss more in detail this problem we recall some definitions and notations of recursion theory (for more details see =-=[12]-=- and [18]). Let p ∈ Q[x] be a polynomial with rational coefficients; the code for p is given by ♯(p). The code for f ∈C[0, 1] ∩ REC is given by a sequence 〈pn : n ∈ N〉 of polynomials pn ∈ Q[x] such th... |

12 |
Functional analysis in weak subsystems of second order arithmetic
- Brown
- 1987
(Show Context)
Citation Context ... . (3) Special case of (2) with X = [0, 1]. Proof. (1) =# (2): Since in ACA 0 every separably closed set in a compact space is closed (see theorem 4.2), the conclusion follows from Brown's result 6.1 =-=[2]. (2)-=- =# (3): Trivial. (3) =# (1): First we prove that (2) implies the following statement which is equivalent to WKL 0 : "Let g 0 , g 1 : N # N be one-toone functions such that #n # N #m # N g 0 (n) ... |

8 |
of second order arithmetic
- Subsystems
- 1999
(Show Context)
Citation Context ...ems. It consists of establishing the weakest subsystem of second order arithmetic in which a theorem of ordinary mathematics can be proved. The basic reference for this program is Simpson's monograph =-=[17]-=- while an overview can be found in [15]. In this paper we carry out a Reverse Mathematics study of the concept of located subsets of a compact complete separable metric space. This concept arises natu... |

5 | Lebesgue numbers and Atsuji spaces in subsystems of second-order arithmetic
- Giusto, Marcone
- 1998
(Show Context)
Citation Context ...A 0 ; (2) closed # located; (3) closed # separably closed; (4) separably closed # closed; (5) separably closed # located; (6) separably closed # weakly located; (7) closed + weakly located # located; =-=(8)-=- closed + weakly located # separably closed. Theorem 1.3. In RCA 0 the following statements are pairwise equivalent: (1) WKL 0 ; (2) closed # weakly located; (3) closed + separably closed # located; (... |

4 |
Π 0 1 classes and boolean combinations of recursively enumerable sets
- Jockusch
(Show Context)
Citation Context ...ets of N are identified with characteristic functions in 2N , P ⊆ 2N ;moreoverPis described by a Π0 1 formula. We equip 2N with the usual product measure µ. Following the argument in Jockusch’s paper =-=[9]-=-, we prove in WWKL0 that the complement of P has measure at most 1/2. Indeed, fixed any e ∈ N,ifA �∈ P, then the measure of the class of such A’s is at most 2−e−2 . Hence the measure of the complement... |

3 |
of closed subsets of a complete separable metric space in weak subsystems of second order arithmetic
- Notions
- 1990
(Show Context)
Citation Context ...re equivalent: (1) ACA 0 . (2) Every closed subset C of a compact complete separable metric space X is located. (3) Every closed set in [0, 1] is located. Proof. (1) =# (2). Since X is compact, Brown =-=[3]-=- (see also theorem 4.2 below) assures that in ACA 0 the notions of closed and separably 12 MARIAGNESE GIUSTO STEPHEN G. SIMPSON closed set coincide. Therefore we may assume that C is a separably close... |

3 |
Measure theory and weak König’s lemma, Archive for Mathematical Logic 30
- Yu, Simpson
- 1990
(Show Context)
Citation Context ...tion of A, where A is the vector space of rational "polynomials" over X under the sup-norm, #f# = sup x#X |f(x)|. Thus C(X) is a separable Banach space. For the precise definitions within RC=-=A 0 , see [20]-=- and Brown's thesis [2, section III.E]. The construction of C(X) within RCA 0 is inspired by the constructive Stone--Weierstrass theorem in the work by Bishop and Bridges [1, 6 MARIAGNESE GIUSTO STEPH... |

2 |
set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary di#erential equations
- Which
- 1984
(Show Context)
Citation Context ... equivalent to WKL 0 : "Let g 0 , g 1 : N # N be one-toone functions such that #n # N #m # N g 0 (n) #= g 1 (m). Then 28 MARIAGNESE GIUSTO STEPHEN G. SIMPSON #Y #m(g 0 (m) # Y # g 1 (m) ## Y ).&q=-=uot; (see [17, 1-=-6]). Let g 0 , g 1 : N # N be one-to-one functions such that #n # N #m # N g 0 (n) #= g 1 (m); we define in [0,1] the separably closed set C = 2 -g 0 (n) : n # N # 2 -g 1 (n) : n # N # {0}. If x # C... |

1 |
Topology and Analysis
- Giusto
- 1998
(Show Context)
Citation Context ...a C(X)-version of Urysohn's lemma (about the separation of two disjoint closed sets by a uniformly continuous function with modulus of uniform continuity) since such a version implies WKL 0 (see e.g. =-=[7]-=- or [14]) and therefore it is not available in RCA 0 . Theorem 6.4 (RCA 0 ). Let X = b A be a compact complete separable metric space, C # X a weakly located closed subset, f : C # R a uniformly conti... |

1 |
1 classes and Boolean combinations of recursively enumerable sets
- Jockusch
- 1974
(Show Context)
Citation Context ... N are identified with characteristic functions in 2 N , P # 2 N ; moreover P is described by a # 0 1 formula. We equip 2 N with the usual product measure . Following the argument in Jockusch's paper =-=[9]-=-, we prove in WWKL 0 that the complement of P has measure at most 1/2. Indeed, fixed any e # N, if A ## P , then the measure of the class of such A's is at most 2 -e-2 . Hence the measure of the compl... |

1 |
of functions with no fixed points
- Degrees
- 1989
(Show Context)
Citation Context ...# A. Hence A ## P . 36 MARIAGNESE GIUSTO STEPHEN G. SIMPSON To every e#ectively immune set A (which is infinite) we can associate a function which is in DNR. The following argument is due to Jockusch =-=[10]-=-. Let g # T A be such that W g(e) = ( the first p(# e (e)) elements of A if # e (e) is defined, # if # e (e) is undefined. We claim that g is a DNR function. If this is not the case, assume that g(e) ... |

1 |
Theory of Recursive Functions and E#ective
- Rogers
(Show Context)
Citation Context ...rem in section 1. On the other hand, to prove that (5) =# (1) is not so easy. To discuss more in detail this problem we recall some definitions and notations of recursion theory (for more details see =-=[12]-=- and [18]). Let p # Q[x] be a polynomial with rational coe#cients; the code for p is given by #(p). The code for f # C[0, 1] # REC is given by a sequence #p n : n # N# of polynomials p n # Q[x] such t... |

1 |
Finite and countable additivity, preprint
- Simpson
- 1996
(Show Context)
Citation Context ...version of Urysohn's lemma (about the separation of two disjoint closed sets by a uniformly continuous function with modulus of uniform continuity) since such a version implies WKL 0 (see e.g. [7] or =-=[14]-=-) and therefore it is not available in RCA 0 . Theorem 6.4 (RCA 0 ). Let X = b A be a compact complete separable metric space, C # X a weakly located closed subset, f : C # R a uniformly continuous fu... |

1 |
of Z 2 and reverse mathematics
- Subsystems
- 1978
(Show Context)
Citation Context ...akest subsystem of second order arithmetic in which a theorem of ordinary mathematics can be proved. The basic reference for this program is Simpson's monograph [17] while an overview can be found in =-=[15]-=-. In this paper we carry out a Reverse Mathematics study of the concept of located subsets of a compact complete separable metric space. This concept arises naturally in the context of metric spaces. ... |