×

zbMATH — the first resource for mathematics

Double convergence and products of Fréchet spaces. (English) Zbl 0954.46002
The paper is devoted to convergence of double sequences and its application to products. In a convergence space three types of double convergences and points, respectively, are recognized; examples are given and their structure and properties are described. The relationship between the topological and convergence closure product of two Fréchet spaces is investigated. In particular, necessary and sufficient condition is given for the topological product of two compact Hausdorff Fréchet spaces to be a Fréchet space.
Reviewer: A.Kufner (Praha)

MSC:
46A19 Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.)
46A04 Locally convex Fréchet spaces and (DF)-spaces
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Arhangel’ski\?, A. V.: The frequency spectrum of a topological space and the product operation. Trudy Moskovs. Mat. Obshch. 40 (1979), 171-206. · Zbl 0447.54004
[2] Boehme, T. K. and Rosenfeld, M.: An example of two compact Fréchet Hausdorff spaces, whose product is not Fréchet. J. London Math. Soc. 8 (1974), 339-344. · Zbl 0289.54026
[3] Frič, R. and Vojtáš, P.: Diagonal conditions in sequential convergence. Convergence Structures 1984 (Proc. Conf. on Convergence, Bechyně, 1984), Akademie-Verlag, Mathematical Research/ Mathematische Forschung Bd. 24, Berlin, 1985, pp. 77-94.
[4] Kendrick, C. T.: On products of Fréchet spaces. Math. Nachr. 65 (1975), 117-123. · Zbl 0295.54008
[5] Kent, D. C.: Decisive convergence spaces, sequential spaces, and Fréchet spaces. Rocky Mtn. J. Math. 1 (1971), 367-374. · Zbl 0218.54015
[6] Koutník, V.: Closure and topological sequential convergence. Convergence Structures 1984 (Proc. Conf. on Convergence, Bechyně, 1984), Akademie-Verlag, Mathematical Research/Mathematische Forschung Bd. 24, Berlin, 1985, pp. 199-204.
[7] Kratochvíl, P.: Multisequences and measure. Gnereal Topology and its Relations to Modern Analysis and Algebra, IV (Proc. Fourth Prague Topological Sympos., 1976) Part B Contributed Papers, Society of Czechoslovak Mathematicians and Physicists, Praha 1977, pp. 237-244.
[8] Michael, E.: A quintuple quotient quest. Gen. Top. Appl. 2. (1972), 91-138. · Zbl 0238.54009
[9] Novák, J.: On some topological spaces represented by systems of sets. Topology and its Applications (Proc. Int. Symposium, Herceg-Novi, 1968). Beograd, 1969, pp. 269-270.
[10] Novák, J.: Concerning the topological product of two Fréchet spaces. Gnereal Topology and its Relations to Modern Analysis and Algebra, IV (Proc. Fourth Prague Topological Sympos., 1976), Part B Contributed Papers, Society of Czechoslovak Mathematicians and Physicists, Praha 1977, pp. 342-343.
[11] Novák, J.: Convergence of double sequences. Convergence Structures 1984 (Proc. Conf. on Convergence, Bechyně, 1984), Akademie-Verlag, Mathematical Research/ Mathematische Forschung Bd. 24, Berlin, 1985, pp. 233-243.
[12] Simon, P.: A compact Fréchet space whose square is not Fréchet. Comment. Math. Univ. Carolinae 21 (1980), 749-753. · Zbl 0466.54022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.