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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:

54F55 Unicoherence, multicoherence
54C30 Real-valued functions in general topology
41A30 Approximation by other special function classes
54E35 Metric spaces, metrizability
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)

Citations:

Zbl 0867.54022
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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, (Choquet, G.; Godefroy, G.; Rogalski, M.; Saint Raymond, J., Publications Mathematiques de l’Univ. Pierre et Marie Curie (1993/1994), Seminaire d’Initiation a l’Analyse, 33ème Année), 7.01-7.09
[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; 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, 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 \(R^n\), Topology Appl., 114, 311-325 (2001) · Zbl 0976.54014
[13] Ciesielski, K.; Shelah, S., A model with no magic set, J. Symbolic Logic, 64, 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; 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
[15] Dikranjan, D., Connectedness and disconnectedness in pseudocompact groups, Rend. Accad. Naz. Sci. XL, Mem. Mat., 110, 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, 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. Categorical Structure of Closure operators with Applications to Topology, Algebra and Discrete Mathematics, Mathematics and Its Applications, 346 (1995), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0853.18002
[21] van Douwen, E., The integers and topology, (Kunen, K.; Vaughan, J. E., Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 111-167
[22] Engelking, R., General Topology. General Topology, Sigma Ser. Pure Math., 6 (1989), Heldermann: Heldermann Berlin
[23] V. Gutev, T. Nogura, Vietoris continuous selections and disconnectedness-like properties, Proc. Amer. Math. Soc., to appear; 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), Springer: Springer Berlin
[25] Hocking, J. G.; Young, G. S., Topology (1988), Dover: Dover New York, (originally published: Addison-Wesley, Reading, MA, 1961)
[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. Uniform Spaces, Mathematical Surveys, 12 (1964), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0124.15601
[28] Knaster, B.; Kuratowski, K., Sur les ensembles connexes, Fund. Math., 2, 206-255 (1921) · JFM 48.0209.02
[29] Kuratowski, K., Topology, Vol. 2 (1968), Academic Press: Academic Press New York
[30] Ursul, M., An example of a plane group whose quasi-component does not coincide with its component, Mat. Zametki, 38, 4, 517-522 (1985)
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