# Article

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Keywords:
sequentially continuous; dyadic compactum; topological group; sequential leader; real-valued measurable cardinal; completion-regular measure
Summary:
We study conditions under which sequentially continuous functions on topological spaces and sequentially continuous homomorphisms of topological groups are continuous.
References:
[A1] Arhangel'skii A.V.: The frequency spectrum of a topological space and the product operation (in Russian). Trudy Mosk. Mat. Ob-va 40 (1979), 171-206; English transl.: Trans. Moscow Math. Soc. 40.2 (1981) 163-200. MR 0550259
[A2] Arhangel'skii A.V.: On countably compact topologies on compact groups and on dyadic compacta. Top. and Appl. 57 (1994), 163-181. MR 1278022
[A3] Arhangel'skii A.V.: On invariants of character and weight type (in Russian). Trudy Mosk. Mat. Ob-va 38 (1979), 3-23; English transl.: Trans. Moscow Math. Soc. 38.2 (1980), 1-23. MR 0544935
[AC] AntonovskiĭM., ChudnovskiĭD.: Some questions of general topology and Tichonov semifields II. Russian Math. Surveys 31 (1976), 69-128.
[B] Booth D.: A Boolean view of sequential compactness. Fund. Math. 85 (1974), 99-102. MR 0367926 | Zbl 0297.54020
[C] Ciesielski K.: Real-valued sequentially continuous functions on the product space $2^\kappa$. unpublished preprint.
[CR] Comfort W.W., Remus D.: Compact groups of Ulam-measurable cardinality: Partial converses of theorems of Arhangel'skii and Varopoulos. Math. Japonica 39.2 (1994), 203-210. MR 1270627
[vD] van Douwen E.K.: The integers and topology. in: Handbook of Set-theoretical Topology, K. Kunen and J.E. Vaughan eds., North Holland, 1984, pp.111-167. MR 0776619 | Zbl 0561.54004
[E] Engelking R.: General Topology. Heldermann Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[E1] Engelking R.: Cartesian products and dyadic spaces. Fund. Math. 57.3 (1965), 287-304. MR 0196692 | Zbl 0173.50603
[EK] Engelking R., Karłowicz M.: Some theorems of set theory and their topological consequences. Fund. Math. 57.3 (1965), 275-285. MR 0196693
[F1] Fremlin D.H.: Measure algebras. in: Handbook of Boolean Algebras, J.D. Monk ed., North Holland, 1989, pp.876-980. MR 0991611 | Zbl 1165.28002
[F2] Fremlin D.H.: Real-valued-measurable cardinals. in: Set Theory of the Reals, H. Judah ed., Israel Mathematical Conference Proceedings 6 (1993) 151-304. MR 1234282 | Zbl 0839.03038
[Gr] Gryllakis C.: Products of completion-regular measures. Proc. Amer. Math. Soc. 103 (1988), 563-568. MR 0943085 | Zbl 0655.28005
[H] Hagler J.: On the structure of $S$ and $C(S)$ for $S$ dyadic. Trans. AMS 213 (1975), 415-428. MR 0388062 | Zbl 0321.46022
[Hal] Halmos P.R.: Measure Theory. Van Nostrand, New York, 1950. MR 0033869 | Zbl 0283.28001
[Hu] Hušek M.: Sequentially continuous homomorphisms on products of topological groups. preprint, October 1995. MR 1397074
[J] Jech T.: Set Theory. Academic Press, 1978. MR 0506523 | Zbl 1007.03002
[Iv] IvanovskĭL.N.: On a hypothesis of P.S. Alexandroff. Dokl. AN SSSR 123.4 (1958), 785-786.
[Ku] Kuz'minov V.I.: On a hypothesis of P.S. Alexandroff in the theory of topological groups. Dokl. AN SSSR 125.4 (1959), 727-729. MR 0104753
[M] Mazur S.: On continuous mappings on Cartesian products. Fund. Math. 39 (1952), 229-238. MR 0055663
[P1] Plebanek G.: On the space of continuous function on a dyadic set. Mathematika 38 (1991), 42-49. MR 1116683
[P2] Plebanek G.: Remarks on measurable Boolean algebras and sequential cardinals. Fund. Math. 143 (1993), 11-22. MR 1234988 | Zbl 0788.28003
[Pas] Pasynkov B.A.: Sections over zero-dimensional subsets of quotients of locally compact groups (in Russian). Dokl. AN SSSR 178.6 (1968), 1255-1258; English translation in: Soviet Math. Dokl. 9.1 (1968), 281-284. MR 0225925
[Pn] Pontryagin L.S.: Topological Groups. Princeton Univ. Press, Princeton, NJ, 1939. Zbl 0882.01025
[RD] Roelcke W., Dierolf S.: Uniform Structures on Topological Groups and Their Quotients. McGraw-Hill, New York, 1981. MR 0644485 | Zbl 0489.22001
[U] Uspenskiĭ V.V.: Real-valued measurable cardinals and sequentially continuous homomorphisms. preprint, December 1993. MR 1234282
[Vn] Vaughan J.E.: Countably compact spaces and sequentially compact spaces. in: Handbook of Set-theoretical Topology, K. Kunen and J.E. Vaughan eds., North Holland, 1984, pp.569-602. MR 0776631
[Vs] Varopoulos N.T.: A theorem on the continuity of homomorphisms of locally compact groups. Proc. Cambridge Phil. Soc. 60 (1964), 449-463. MR 0162880 | Zbl 0121.03704
[Wh] Wheeler R.F.: A survey on Baire measures and strict topologies. Exp. Math. 2 (1983), 97-190. MR 0710569

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