zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Uniform quasi components, thin spaces and compact separation. (English) Zbl 1067.54027
Summary: We prove that every complete metric space $X$ that is thin (i.e., every closed subspace has connected uniform quasi components) has the compact separation property (for any two disjoint closed connected subspaces $A$ and $B$ of $X$ there is a compact set $K$ disjoint from $A$ and $B$ such that every neighbourhood of $K$ disjoint from $A$ and $B$ separates $A$ and $B$). The real line and all compact spaces are obviously thin. We show that a space is thin if and only if it does not contain a certain forbidden configuration. Finally, we prove that every metric $UA$-space [see the first two authors, Rend. Inst. Mat. Univ. Trieste 25, 23--55 (1993; Zbl 0867.54022)] is thin. The $UA$-spaces form a class properly including the Atsuji spaces.

MSC:
54F55Unicoherence, multicoherence
54C30Real-valued functions on topological spaces
41A30Approximation by other special function classes
54E35Metric spaces, metrizability
54D15Higher separation axioms
WorldCat.org
Full Text: DOI
References:
[1] Atsuji, M.: Uniform continuity of continuous functions on metric spaces. Pacific J. Math. 8, 11-16 (1958) · Zbl 0082.16207
[2] Atsuji, M.: Uniform continuity of continuous functions on uniform spaces. Canad. J. Math. 13, 657-663 (1961) · Zbl 0102.37703
[3] Berarducci, A.; Dikranjan, D.: Uniformly approachable functions and UA spaces. Rend. istit. Mat. univ. Trieste 25, 23-56 (1993) · Zbl 0867.54022
[4] Berarducci, A.; Dikranjan, D.: Uniformly approachable functions. Publications mathematiques de l’univ. Pierre et marie Curie, 7.01-7.09 (1993/1994)
[5] Berarducci, A.; Dikranjan, D.; Pelant, J.: Functions with distant fibers and uniform continuity. Topology appl. 121, 3-23 (2002) · Zbl 1009.54019
[6] A. Berarducci, D. Dikranjan, J. Pelant, Uniformly approachable spaces. II, Work in progress · Zbl 1078.54014
[7] Burke, M.: Characterizing uniform continuity with closure operations. Topology appl. 59, 245-259 (1994) · Zbl 0847.54002
[8] Burke, M.: Continuous functions which take a somewhere dense set of values on every open set. Topology appl. 103, No. 1, 95-110 (2000) · Zbl 0958.54009
[9] Burke, M.; Ciesielski, K.: Sets on which measurable functions are determined by their range. Canad. math. J. 49, 1089-1116 (1997) · Zbl 0905.28001
[10] Burke, M.; Ciesielski, K.: Sets of range uniqueness for classes of continuous functions. Proc. amer. Math. soc. 127, 3295-3304 (1999) · Zbl 0939.26003
[11] Ciesielski, K.; Dikranjan, D.: Uniformly approachable maps. Topology proc. 20, 75-89 (1995) · Zbl 0899.54021
[12] Ciesielski, K.; Dikranjan, D.: Between continuous and uniformly continuous functions on rn. Topology appl. 114, 311-325 (2001) · Zbl 0976.54014
[13] Ciesielski, K.; Shelah, S.: A model with no magic set. J. symbolic logic 64, No. 4, 1467-1490 (1999) · Zbl 0945.03074
[14] E. Cuchillo-Ibáñez, M. Morón, F. Ruiz del Portal, Closed mappings and spaces in which components and quasicomponents coincide (in Spanish), Mathematical Contributions, Editorial Complutense, Madrid, 1994, pp. 357--363 · Zbl 0816.54007
[15] Dikranjan, D.: Connectedness and disconnectedness in pseudocompact groups. Rend. accad. Naz. sci. XL, mem. Mat. 110, No. XVI (12), 211-221 (1992) · Zbl 0834.22005
[16] Dikranjan, D.: Dimension and connectedness in pseudo-compact groups. Comp. rend. Acad. sci. Paris, sér. I 316, 309-314 (1993) · Zbl 0783.54029
[17] Dikranjan, D.: Zero-dimensionality of some pseudocompact groups. Proc. amer. Math. soc. 120, No. 4, 1299-1308 (1994) · Zbl 0846.54027
[18] Dikranjan, D.: Compactness and connectedness in topological groups. Topology appl. 84, 227-252 (1998) · Zbl 0995.54036
[19] Dikranjan, D.; Pelant, J.: The impact of closure operators on the structure of a concrete category. Quaestiones math. 18, 381-396 (1995) · Zbl 0864.54024
[20] Dikranjan, D.; Tholen, W.: Categorical structure of closure operators with applications to topology, algebra and discrete mathematics. Mathematics and its applications 346 (1995) · Zbl 0853.18002
[21] Van Douwen, E.: The integers and topology. Handbook of set-theoretic topology, 111-167 (1984)
[22] Engelking, R.: General topology. Sigma ser. Pure math. 6 (1989) · Zbl 0684.54001
[23] V. Gutev, T. Nogura, Vietoris continuous selections and disconnectedness-like properties, Proc. Amer. Math. Soc., to appear · Zbl 0973.54021
[24] Hewitt, E.; Ross, K.: Abstract harmonic analysis, vol. I. (1979) · Zbl 0416.43001
[25] Hocking, J. G.; Young, G. S.: Topology. (1988) · Zbl 0718.55001
[26] Hueber, H.: On uniform continuity and compactness in metric spaces. Amer. math. Monthly 88, 204-205 (1981) · Zbl 0451.54024
[27] Isbell, J. R.: Uniform spaces. Mathematical surveys 12 (1964) · Zbl 0124.15601
[28] Knaster, B.; Kuratowski, K.: Sur LES ensembles connexes. Fund. math. 2, 206-255 (1921) · Zbl 48.0209.02
[29] Kuratowski, K.: Topology, vol. 2. (1968)
[30] Ursul, M.: An example of a plane group whose quasi-component does not coincide with its component. Mat. zametki 38, No. 4, 517-522 (1985) · Zbl 0598.22005