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Uspekhi Mat. Nauk, 1976, Volume 31, Issue 5(191), Pages 17–32 (Mi umn3840)  

This article is cited in 18 scientific papers (total in 18 papers)

On some topological spaces that occur in functional analysis

A. V. Arkhangel'skii

Abstract: This is a study of the topological properties of spaces of continuous real functions on compact sets in the topology of pointwise convergence. Compact subsets in these spaces are called functionally complete. The topological properties of functionally complete compacta are established, among them the fact that the density of each subspace of a functionally complete compactum is equal to its weight. Compacta of countable tightness having this last property are called exact. Each functionally complete compactum is exact. It is proved that each exact compactum is a Fréchet–Uryson space and satisfies the first axiom of countability on an everywhere dense set of points. The continuous image of an exact compactum is exact. Recently M. Vage has constructed a “naive” example of an exact but not functionally complete compact space. Another interesting question is: does there exist a non-metrizable, homogeneous, functionally complete compactum?

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English version:
Russian Mathematical Surveys, 1976, 31:5, 14–30

Bibliographic databases:

UDC: 513.831
MSC: 46A19, 46A50
Received: 27.05.1976

Citation: A. V. Arkhangel'skii, “On some topological spaces that occur in functional analysis”, Uspekhi Mat. Nauk, 31:5(191) (1976), 17–32; Russian Math. Surveys, 31:5 (1976), 14–30

Citation in format AMSBIB
\by A.~V.~Arkhangel'skii
\paper On some topological spaces that occur in functional analysis
\jour Uspekhi Mat. Nauk
\yr 1976
\vol 31
\issue 5(191)
\pages 17--32
\jour Russian Math. Surveys
\yr 1976
\vol 31
\issue 5
\pages 14--30

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    This publication is cited in the following articles:
    1. Yu. A. Abramovich, “On a new class of Fréchet–Urysohn spaces”, Russian Math. Surveys, 33:5 (1978), 177–178  mathnet  crossref  mathscinet  zmath
    2. A. V. Arkhangel'skii, “Structure and classification of topological spaces and cardinal invariants”, Russian Math. Surveys, 33:6 (1978), 33–96  mathnet  crossref  mathscinet  zmath
    3. S. P. Gul'ko, “On the structure of spaces of continuous functions and their complete paracompactness”, Russian Math. Surveys, 34:6 (1979), 36–44  mathnet  crossref  mathscinet  zmath
    4. Mohammad Ismail, Peter Nyikos, “On spaces in which countably compact sets are closed, and hereditary properties”, Topology and its Applications, 11:3 (1980), 281  crossref
    5. E. G. Pytkeev, “On the tightness of spaces of continuous functions”, Russian Math. Surveys, 37:1 (1982), 176–177  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. A. V. Arkhangel'skii, “Function spaces in the topology of pointwise convergence, and compact sets”, Russian Math. Surveys, 39:5 (1984), 9–56  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    7. A. V. Arkhangel'skii, “Topological homogeneity. Topological groups and their continuous images”, Russian Math. Surveys, 42:2 (1987), 83–131  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    8. O.G. Okunev, “On Lindelöf Σ-spaces of continuous functions in the pointwise topology”, Topology and its Applications, 49:2 (1993), 149  crossref
    9. Toshihiro Nagamizu, “A note on fragmentable topological spaces”, BAZ, 49:1 (1994), 91  crossref  mathscinet  zmath  isi
    10. M. Bell, “The hyperspace of a compact space, I”, Topology and its Applications, 72:1 (1996), 39  crossref
    11. Henno Brandsma, Jan van Mill, “Monotone normality, measures and hyperspaces”, Topology and its Applications, 85:1-3 (1998), 287  crossref
    12. A.V. Arhangel'skii, “Some observations on Cp-theory and bibliography”, Topology and its Applications, 89:3 (1998), 203  crossref
    13. H.J.K. Junnila, “Embeddings of Weakly Compact Sets and *-Paired Banach Spaces”, Journal of Functional Analysis, 177:2 (2000), 442  crossref
    14. Oleg Okunev, Angel Tamariz-Mascarúa, “On the Čech number of Cp(X)”, Topology and its Applications, 137:1-3 (2004), 237  crossref
    15. Montserrat Bruguera, Mikhail Tkachenko, “The three space problem in topological groups”, Topology and its Applications, 153:13 (2006), 2278  crossref
    16. María Muñoz, “A note on the theorem of Baturov”, BAZ, 76:2 (2007), 219  crossref  mathscinet  zmath  isi
    17. A. Dow, H. Junnila, J. Pelant, “Chain conditions and weak topologies”, Topology and its Applications, 156:7 (2009), 1327  crossref
    18. B. Cascales, M. Muñoz, J. Orihuela, “The number of K-determination of topological spaces”, RACSAM, 2012  crossref
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