Geodesic
Weak continuity properties of topologized groups
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 1, pp. 133-148 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We explore (weak) continuity properties of group operations. For this purpose, the Novak number and developability number are applied. It is shown that if $(G, \cdot ,\tau )$ is a regular right (left) semitopological group with $\mathop{{\rm dev}}(G)\mathop{{\rm Nov}}(G)$ such that all left (right) translations are feebly continuous, then $(G,\cdot ,\tau )$ is a topological group. This extends several results in literature.
We explore (weak) continuity properties of group operations. For this purpose, the Novak number and developability number are applied. It is shown that if $(G, \cdot ,\tau )$ is a regular right (left) semitopological group with $\mathop{{\rm dev}}(G)\mathop{{\rm Nov}}(G)$ such that all left (right) translations are feebly continuous, then $(G,\cdot ,\tau )$ is a topological group. This extends several results in literature.
Classification : 22A05, 54C08, 54E52, 54H11
Keywords: developability number; feebly continuous; nearly continuous; Novak number; paratopological group; semitopological group; topological group 
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Cao, J.; Drozdowski, R.; Piotrowski, Z. Weak continuity properties of topologized groups. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 1, pp. 133-148. http://geodesic.mathdoc.fr/item/CMJ_2010_60_1_a11/

[1] Andrijevi'c, D.: Semi-preopen sets. Mat. Ves. 38 (1986), 24-32.

[2] Arhangel'skii, A. V.: Mappings and spaces. Russ. Math. Surv. 21 (1966), 115-162. | MR

[3] Arhangel'skii, A. V., Reznichenko, E. A.: Paratopological and semitopological groups versus topological groups. Topology Appl. 151 (2005), 107-119. | DOI | MR | Zbl

[4] Banakh, T., Ravsky, O.: Oscillator topologies on a paratopological group and related number invariants. Algebraic Structures and Their Applications. Proc. Third International Algebraic Conference, Kiev, Ukraine, July 2-8, 2001 Instytut Matematyky NAN Kiev (2002), 140-153. | MR | Zbl

[5] Banakh, T., Ravsky, S.: On subgroups of saturated or totally bounded paratopological groups. Algebra Discrete Math. (2003), 1-20. | MR | Zbl

[6] Bella, A.: Some remarks on the Novak number. General topology and its relations to modern analysis and algebra VI (Prague, 1986) Heldermann Berlin (1988), 43-48. | MR | Zbl

[7] Bohn, E., Lee, J.: Semi-topological groups. Am. Math. Mon. 72 (1965), 996-998. | DOI | MR | Zbl

[8] Bourbaki, N.: Elements of Mathematics, General Topology, Chapters 1-4. Springer Berlin (1989). | MR | Zbl

[9] Bouziad, A.: The Ellis theorem and continuity in group. Topology Appl. 50 (1993), 73-80. | DOI | MR

[10] Bouziad, A.: Continuity of separately continuous group actions in $p$-spaces. Topology Appl. 71 (1996), 119-124. | DOI | MR | Zbl

[11] Cao, J., Greenwood, S.: The ideal generated by $\sigma$-nowhere dense sets. Appl. Gen. Topol. 7 (2006), 253-264. | DOI | MR | Zbl

[12] Engelking, R.: General Topology. Revised and completed edition. Heldermann-Verlag Berlin (1989). | MR

[13] Ferri, S., Hernández, S., Wu, T. S.: Continuity in topological groups. Topology Appl. 153 (2006), 1451-1457. | DOI | MR

[14] Frolík, Z.: Remarks concerning the invariance of Baire spaces under mappings. Czechoslovak Math. J. 11 (1961), 381-385. | MR

[15] Gentry, K. R., Hoyle, H. B.: Somewhat continuous functions. Czechoslovak Math. J. 21 (1971), 5-12. | MR | Zbl

[16] Guran, I.: Cardinal invariants of paratopological grups. 2nd International Algebraic Conference in Ukraine Vinnytsia (1999).

[17] J. L. Kelley, I. Namioka, W. F. Donoghue jun., K. R. Lucas, B. J. Pettis, T. E. Poulsen, G. B. Price, W. Robertson, W. R. Scott, K. T. Smith: Linear Topological Spaces. D. Van Nostarand Company, Inc. Princeton (1963). | MR

[18] Kempisty, S.: Sur les fonctions quasicontinues. Fundam. Math. 19 (1932), 184-197 French. | DOI | Zbl

[19] Kenderov, P. S., Kortezov, I. S., Moors, W. B.: Topological games and topological groups. Topology Appl. 109 (2001), 157-165. | DOI | MR | Zbl

[20] Lau, A. T.-M., Loy, R. J.: Banach algebras on compact right topological groups. J. Funct. Anal. 225 (2005), 263-300. | DOI | MR | Zbl

[21] Liu, C.: A note on paratopological group. Commentat. Math. Univ. Carol. 47 (2006), 633-640. | MR

[22] Mercourakis, S., Negrepontis, S.: Banach Spaces and Topology. II. Recent Progress in General Topology (Prague, 1991). North-Holland Amsterdam (1992), 493-536. | MR

[23] Montgomery, D.: Continuity in topological groups. Bull. Am. Math. Soc. 42 (1936), 879-882. | DOI | MR | Zbl

[24] Neubrunn, T.: A generalized continuity and product spaces. Math. Slovaca 26 (1976), 97-99. | MR | Zbl

[25] Neubrunn, T.: Quasi-continuity. Real Anal. Exch. 14 (1989), 259-306. | DOI | MR | Zbl

[26] Piotrowski, Z.: Quasi-continuity and product spaces. Proc. Int. Conf. on Geometric Topology, Warszawa 1978 PWN Warsaw (1980), 349-352. | MR | Zbl

[27] Piotrowski, Z.: Separate and joint continuity. Real Anal. Exch. 11 (1985-86), 293-322. | DOI | MR | Zbl

[28] Piotrowski, Z.: Separate and joint continuity II. Real Anal. Exch. 15 (1990), 248-258. | DOI | MR | Zbl

[29] Piotrowski, Z.: Separate and joint continuity in Baire groups. Tatra Mt. Math. Publ. 14 (1998), 109-116. | MR | Zbl

[30] Pták, V.: Completeness and the open mapping theorem. Bull. Soc. Math. Fr. 86 (1958), 41-74. | DOI | MR

[31] Ravsky, O.: Paratopological groups. II. Math. Stud. 17 (2002), 93-101. | MR | Zbl

[32] Rothmann, D. D.: A nearly discrete metric. Am. Math. Mon. 81 (1974), 1018-1019. | DOI | MR | Zbl

[33] Ruppert, W.: Compact Semitopological Semigroups: An Intrinsic Theory. Lecture Notes in Mathematics Vol. 1079. Springer (1984). | MR

[34] Solecki, S., Srivastava, S. M.: Automatic continuity of group operations. Topology Appl. 77 (1997), 65-75. | DOI | MR | Zbl

[35] Talagrand, M.: Espaces de Baire et espaces de Namioka. Math. Ann. 270 (1985), 159-164 French. | DOI | MR | Zbl

[36] Tkachenko, M.: Paratopological groups versus topological groups. Lecture at Advances in Set-Theoretic Topology. Conference in Honour of Tsugunori Nogura on his 60th Birthday, Erice, June 2008.

[37] Zelazko, W.: A theorem on $B_0$ division algebras. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 8 (1960), 373-375. | MR | Zbl

[38] Zelazko, W.: A theorem on $B_0$ division algebras. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 8 (1960), 373-375. | MR | Zbl