Previous |  Up |  Next

Article

Keywords:
developability number; feebly continuous; nearly continuous; Novak number; paratopological group; semitopological group; topological group
Summary:
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.
References:
[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 0227950
[3] Arhangel'skii, A. V., Reznichenko, E. A.: Paratopological and semitopological groups versus topological groups. Topology Appl. 151 (2005), 107-119. DOI 10.1016/j.topol.2003.08.035 | MR 2139745 | Zbl 1077.54023
[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 2210489 | Zbl 1098.22004
[5] Banakh, T., Ravsky, S.: On subgroups of saturated or totally bounded paratopological groups. Algebra Discrete Math. (2003), 1-20. MR 2070399 | Zbl 1061.22003
[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 0952589 | Zbl 0634.54004
[7] Bohn, E., Lee, J.: Semi-topological groups. Am. Math. Mon. 72 (1965), 996-998. DOI 10.2307/2313342 | MR 0190259 | Zbl 0134.03601
[8] Bourbaki, N.: Elements of Mathematics, General Topology, Chapters 1-4. Springer Berlin (1989). MR 0979294 | Zbl 0683.54003
[9] Bouziad, A.: The Ellis theorem and continuity in group. Topology Appl. 50 (1993), 73-80. DOI 10.1016/0166-8641(93)90074-N | MR 1217698
[10] Bouziad, A.: Continuity of separately continuous group actions in $p$-spaces. Topology Appl. 71 (1996), 119-124. DOI 10.1016/0166-8641(95)00039-9 | MR 1399551 | Zbl 0855.22006
[11] Cao, J., Greenwood, S.: The ideal generated by $\sigma$-nowhere dense sets. Appl. Gen. Topol. 7 (2006), 253-264. DOI 10.4995/agt.2006.1928 | MR 2295174 | Zbl 1114.54021
[12] Engelking, R.: General Topology. Revised and completed edition. Heldermann-Verlag Berlin (1989). MR 1039321
[13] Ferri, S., Hernández, S., Wu, T. S.: Continuity in topological groups. Topology Appl. 153 (2006), 1451-1457. DOI 10.1016/j.topol.2005.04.007 | MR 2211210
[14] Frolík, Z.: Remarks concerning the invariance of Baire spaces under mappings. Czechoslovak Math. J. 11 (1961), 381-385. MR 0133098
[15] Gentry, K. R., Hoyle, H. B.: Somewhat continuous functions. Czechoslovak Math. J. 21 (1971), 5-12. MR 0278269 | Zbl 0222.54010
[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 0166578
[18] Kempisty, S.: Sur les fonctions quasicontinues. Fundam. Math. 19 (1932), 184-197 French. Zbl 0005.19802
[19] Kenderov, P. S., Kortezov, I. S., Moors, W. B.: Topological games and topological groups. Topology Appl. 109 (2001), 157-165. DOI 10.1016/S0166-8641(99)00152-2 | MR 1806330 | Zbl 0976.22003
[20] Lau, A. T.-M., Loy, R. J.: Banach algebras on compact right topological groups. J. Funct. Anal. 225 (2005), 263-300. DOI 10.1016/j.jfa.2005.04.006 | MR 2152500 | Zbl 1098.46035
[21] Liu, C.: A note on paratopological group. Commentat. Math. Univ. Carol. 47 (2006), 633-640. MR 2337418
[22] Mercourakis, S., Negrepontis, S.: Banach Spaces and Topology. II. Recent Progress in General Topology (Prague, 1991). North-Holland Amsterdam (1992), 493-536. MR 1229137
[23] Montgomery, D.: Continuity in topological groups. Bull. Am. Math. Soc. 42 (1936), 879-882. DOI 10.1090/S0002-9904-1936-06456-6 | MR 1563458 | Zbl 0015.39403
[24] Neubrunn, T.: A generalized continuity and product spaces. Math. Slovaca 26 (1976), 97-99. MR 0436064 | Zbl 0318.54008
[25] Neubrunn, T.: Quasi-continuity. Real Anal. Exch. 14 (1989), 259-306. MR 0995972 | Zbl 0679.26003
[26] Piotrowski, Z.: Quasi-continuity and product spaces. Proc. Int. Conf. on Geometric Topology, Warszawa 1978 PWN Warsaw (1980), 349-352. MR 0656769 | Zbl 0481.54007
[27] Piotrowski, Z.: Separate and joint continuity. Real Anal. Exch. 11 (1985-86), 293-322. MR 0844254 | Zbl 0606.54009
[28] Piotrowski, Z.: Separate and joint continuity II. Real Anal. Exch. 15 (1990), 248-258. MR 1042540 | Zbl 0702.54009
[29] Piotrowski, Z.: Separate and joint continuity in Baire groups. Tatra Mt. Math. Publ. 14 (1998), 109-116. MR 1651201 | Zbl 0938.22001
[30] Pták, V.: Completeness and the open mapping theorem. Bull. Soc. Math. Fr. 86 (1958), 41-74. MR 0105606
[31] Ravsky, O.: Paratopological groups. II. Math. Stud. 17 (2002), 93-101. MR 1932275 | Zbl 1018.22001
[32] Rothmann, D. D.: A nearly discrete metric. Am. Math. Mon. 81 (1974), 1018-1019. DOI 10.2307/2319315 | MR 0350705 | Zbl 0292.26001
[33] Ruppert, W.: Compact Semitopological Semigroups: An Intrinsic Theory. Lecture Notes in Mathematics Vol. 1079. Springer (1984). MR 0762985
[34] Solecki, S., Srivastava, S. M.: Automatic continuity of group operations. Topology Appl. 77 (1997), 65-75. DOI 10.1016/S0166-8641(96)00119-8 | MR 1443429 | Zbl 0882.22001
[35] Talagrand, M.: Espaces de Baire et espaces de Namioka. Math. Ann. 270 (1985), 159-164 French. DOI 10.1007/BF01456180 | MR 0771977 | Zbl 0582.54008
[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 0125901 | Zbl 0095.31303
[38] Zelazko, W.: A theorem on $B_0$ division algebras. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 8 (1960), 373-375. MR 0125901 | Zbl 0095.31303
Partner of
EuDML logo