A theorem about countable decomposability

1982 ◽  
Vol 91 (3) ◽  
pp. 457-458 ◽  
Author(s):  
Roy O. Davies ◽  
Claude Tricot
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Continuous Functions ◽  
Perfect Set ◽  
Baire Function ◽  
Separable Metric Space

A function f:X → ℝ is countably decomposable (into continuous functions) if the topological space X can be partitioned into countably many sets An with each restriction f│ An continuous. According to L. V. Keldysh(2), the question whether every Baire function is countably decomposable was first raised by N. N. Luzin, and answered by P. S. Novikov. The answer is negative even for Baire-1 functions, as is shown in (2) (see also (1). In this paper we develop a characterization of the countably decomposable functions on a separable metric space X (see Corollary 1). We deduce that when X is complete they include all functions possessing the property P defined by D. E. Peek in (3): each non-empty σ-perfect set H contains a point at which f│ H is continuous. The example given by Peek shows that not every countably decomposable Baire-1 function has property P.

scholarly journals Note on limits of simply continuous and cliquish functions

1994 ◽  
Vol 17 (3) ◽  
pp. 447-450 ◽  
Author(s):  
Janina Ewert
Keyword(s):  
Metric Space ◽  
Continuous Functions ◽  
Baire Space ◽  
Baire Property ◽  
Baire Class ◽  
Class 1 ◽  
Pointwise Limit ◽  
Separable Metric Space

The main result of this paper is that any functionfdefined on a perfect Baire space(X,T)with values in a separable metric spaceYis cliquish (has the Baire property) iff it is a uniform (pointwise) limit of sequence{fn:n≥1}of simply continuous functions. This result is obtained by a change of a topology onXand showing that a functionf:(X,T)→Yis cliquish (has the Baire property) iff it is of the Baire class 1 (class 2) with respect to the new topology.

scholarly journals Characterizations of Ideals in IntermediateC-RingsA(X)via theA-Compactifications ofX

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Joshua Sack ◽  
Saleem Watson
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Completely Regular ◽  
Intermediate Ring ◽  
Maximal Ideals ◽  
Kolmogorov Theorem

LetXbe a completely regular topological space. An intermediate ring is a ringA(X)of continuous functions satisfyingC*(X)⊆A(X)⊆C(X). In Redlin and Watson (1987) and in Panman et al. (2012), correspondences𝒵AandℨAare defined between ideals inA(X)andz-filters onX, and it is shown that these extend the well-known correspondences studied separately forC∗(X)andC(X), respectively, to any intermediate ring. Moreover, the inverse map𝒵A←sets up a one-one correspondence between the maximal ideals ofA(X)and thez-ultrafilters onX. In this paper, we define a function𝔎Athat, in the case thatA(X)is aC-ring, describesℨAin terms of extensions of functions to realcompactifications ofX. For such rings, we show thatℨA←mapsz-filters to ideals. We also give a characterization of the maximal ideals inA(X)that generalize the Gelfand-Kolmogorov theorem fromC(X)toA(X).

Distance Sets of Urysohn Metric Spaces

2013 ◽  
Vol 65 (1) ◽  
pp. 222-240 ◽  
Author(s):  
N.W. Sauer
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Finite Metric Space ◽  
Distance Set ◽  
Distance Sets ◽  
Separable Metric Space

Abstract.A metric space M = (M; d) is homogeneous if for every isometry f of a finite subspace of M to a subspace of M there exists an isometry of M onto M extending f . The space M is universal if it isometrically embeds every finite metric space F with dist(F) ⊆ dist(M) (with dist(M) being the set of distances between points in M).A metric space U is a Urysohn metric space if it is homogeneous, universal, separable, and complete. (We deduce as a corollary that a Urysohn metric space U isometrically embeds every separable metric space M with dist(M) ⊆ dist(U).)The main results are: (1) A characterization of the sets dist(U) for Urysohn metric spaces U. (2) If R is the distance set of a Urysohn metric space and M and N are two metric spaces, of any cardinality with distances in R, then they amalgamate disjointly to a metric space with distances in R. (3) The completion of every homogeneous, universal, separable metric space M is homogeneous.

scholarly journals Spaces with σ-locally finite Lindelöf sn-networks

2013 ◽  
Vol 93 (107) ◽  
pp. 145-152
Author(s):  
Luong Tuyen
Keyword(s):  
Metric Space ◽  
Locally Finite ◽  
Weak Bases ◽  
Locally Countable ◽  
Separable Metric Space

We prove that a space X has a ?-locally finite Lindel?f sn-network if and only if X is a compact-covering compact and mssc-image of a locally separable metric space, if and only if X is a sequentially-quotient ? and mssc-image of a locally separable metric space, where ?compact-covering? (or ?sequentially-quotient?) can not be replaced by ?sequence-covering?. As an application, we give a new characterization of spaces with locally countable weak bases.

Reverse Mathematics and Π12 Comprehension

2005 ◽  
Vol 11 (4) ◽  
pp. 526-533 ◽  
Author(s):  
Carl Mummert ◽  
Stephen G. Simpson
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Reverse Mathematics ◽  
Second Order ◽  
Converse Statement ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Complete Separable Metric Space ◽  
Separable Metric Space ◽  
Metrization Theorem

AbstractWe initiate the reverse mathematics of general topology. We show that a certain metrization theorem is equivalent to Π12 comprehension. An MF space is defined to be a topological space of the form MF(P) with the topology generated by {Np ∣ p ϵ P}. Here P is a poset, MF(P) is the set of maximal filters on P, and Np = {F ϵ MF(P) ∣ p ϵ F }. If the poset P is countable, the space MF(P) is said to be countably based. The class of countably based MF spaces can be defined and discussed within the subsystem ACA0 of second order arithmetic. One can prove within ACA0 that every complete separable metric space is homeomorphic to a countably based MF space which is regular. We show that the converse statement, “every countably based MF space which is regular is homeomorphic to a complete separable metric space,” is equivalent to . The equivalence is proved in the weaker system . This is the first example of a theorem of core mathematics which is provable in second order arithmetic and implies Π12 comprehension.

scholarly journals F-Rings of Continuous Functions I

1959 ◽  
Vol 11 ◽  
pp. 80-86 ◽  
Author(s):  
Barron Brainerd
Keyword(s):  
Topological Space ◽  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
The Other ◽  
Topological Spaces ◽  
Closed Sets ◽  
Real Complex

It is well known (2, 4) that the ring of all real (complex) continuous functions on a compact Hausdorff space can be characterized algebraically as a Banach algebra which satisfies certain additional intrinsic conditions. It might be expected that rings of all continuous functions on other topological spaces also have algebraic characterizations. The main purpose of this note is to discuss two such characterizations. In both cases the characterizations are given in the terms of the theory of F-brings (1). In one case a characterization is given for the ring of all (real) continuous functions on a generalized P-space, that is, a zero-dimensional topological space in which the class of open-closed sets forms a σ-algebra. A Hausdorff generalized P-space is a P-space in the terminology of (3). In the other case a theorem of Sikorski (6) is employed to give a characterization of the ring of all (real) continuous functions on an upper X1-compact P-space.

REVERSE MATHEMATICS OF MF SPACES

2006 ◽  
Vol 06 (02) ◽  
pp. 203-232 ◽  
Author(s):  
CARL MUMMERT
Keyword(s):  
Metric Space ◽  
Closed Subset ◽  
Metric Spaces ◽  
Reverse Mathematics ◽  
General Topology ◽  
Second Order Arithmetic ◽  
Perfect Set ◽  
Order Arithmetic ◽  
Complete Separable Metric Space ◽  
Separable Metric Space

This paper gives a formalization of general topology in second-order arithmetic using countably based MF spaces. This formalization is used to study the reverse mathematics of general topology. For each poset P we let MF (P) denote the set of maximal filters on P endowed with the topology generated by {Np | p ∈ P}, where Np = {F ∈ MF (P) | p ∈ F}. We define a countably based MF space to be a space of the form MF (P) for some countable poset P. The class of countably based MF spaces includes all complete separable metric spaces as well as many nonmetrizable spaces. The following reverse mathematics results are obtained. The proposition that every nonempty Gδ subset of a countably based MF space is homeomorphic to a countably based MF space is equivalent to [Formula: see text] over ACA0. The proposition that every uncountable closed subset of a countably based MF space contains a perfect set is equivalent over [Formula: see text] to the proposition that [Formula: see text] is countable for all A ⊆ ℕ. The proposition that every regular countably based MF space is homeomorphic to a complete separable metric space is equivalent to [Formula: see text] over [Formula: see text].

Selectionable distributions for a random set

1990 ◽  
Vol 108 (2) ◽  
pp. 405-408 ◽  
Author(s):  
David Ross
Keyword(s):  
Compact Subset ◽  
Metric Space ◽  
Random Set ◽  
Complete Separable Metric Space ◽  
Separable Metric Space

AbstractArtstein has given a characterization of the distributions induced by selections of a random compact subset of a complete separable metric space. In this paper, Artstein's results are extended to spaces which may be neither metric nor separable.

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