# Article

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Keywords:
inverse system; resolution; Stone-Čech compactification; pseudocompact space; shape; strong shape
Summary:
J. Keesling has shown that for connected spaces $X$ the natural inclusion $e:X\rightarrow \beta X$ of $X$ in its Stone-Čech compactification is a shape equivalence if and only if $X$ is pseudocompact. This paper establishes the analogous result for strong shape. Moreover, pseudocompact spaces are characterized as spaces which admit compact resolutions, which improves a result of I. Lončar.
References:
[1] Dowker C.H.: Mapping theorems for noncompact spaces. Amer. J. Math. 69 (1947), 200-240. MR 0020771
[2] Engelking R.: General Topology. Polish Sci. Publ., Warsaw, 1977. MR 0500780 | Zbl 0684.54001
[3] Glicksberg I.: Stone-Čech compactifications of products. Trans. Amer. Math. Soc. 90 (1959), 369-382. MR 0105667 | Zbl 0089.38702
[4] Keesling J.: Shape theory and the Stone-Čech compactification. in Proc. Internat. Conference on Geometric Topology, Polish Sci. Publ., Warsaw, 1980, pp. 236-243. MR 0656750 | Zbl 0477.54018
[5] Keesling K., Sher R.B.: Shape properties of the Stone-Čech compactification. General Topology and Appl. 9 (1978), 1-8. MR 0478105 | Zbl 0396.54018
[6] Lisica Ju.T., Mardešić S.: Coherent prohomotopy and a strong shape category of topological spaces. in Proc. Internat. Topological Conference (Leningrad 1982), Lecture Notes in Math. 1060, Springer-Verlag, Berlin, 1984, pp. 164-173. MR 0770236
[7] Lisica Ju.T., Mardešić S.: Coherent prohomotopy and strong shape theory. Glasnik Mat. 19 (39) (1984), 335-399. MR 0790021
[8] Lončar I.: Some results on resolution of spaces. Rad Jugoslav. Akad. Znan. Umjetn. Matem. Znan. 428 (6) (1987), 37-49. MR 0960457
[9] Mardešić S.: Approximate polyhedra, resolutions of maps and shape fibrations. Fund. Math. 114 (1981), 53-78. MR 0643305
[10] Mardešić S., Segal J.: Shape Theory. North-Holland, Amsterdam, 1982. MR 0676973
[11] Morita K.: On shapes of topological spaces. Fund. Math. 86 (1975), 251-259. MR 0388385 | Zbl 0296.54034
[12] Walker R.C.: The Stone-Čech compactification. Springer-Verlag, Berlin, 1974. MR 0380698 | Zbl 0292.54001

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