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Transactions of the American Mathematical Society

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Topological spaces with prescribed nonconstant continuous mappings


Author: Věra Trnková
Journal: Trans. Amer. Math. Soc. 261 (1980), 463-482
MSC: Primary 54C05; Secondary 20M20, 54H10
DOI: https://doi.org/10.1090/S0002-9947-1980-0580898-9
MathSciNet review: 580898
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Abstract: Given a $ {T_1}$-space Y and a $ {T_3}$-space V, consider $ {T_3}$-spaces X such that X has a closed covering by spaces homeomorphic to V and any continuous mapping $ f:\,X \to Y$ is constant. All such spaces and all their continuous mappings are shown to form a very comprehensive category, containing, e.g., a proper class of spaces without nonconstant, nonidentical mappings or containing a space X, for every monoid M, such that all the nonconstant continuous mappings of X into itself are closed under composition and form a monoid isomorphic to M. The category of paracompact connected spaces, having a closed covering by a given totally disconnected paracompact space, has, e.g., analogous properties. Categories of metrizable spaces are also investigated.


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  • [1] H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1967), 241-249. MR 0220249 (36:3315)
  • [2] T. E. Gantner, A regular space on which every continuous real-valued function is constant, Amer. Math. Monthly 78 (1971), 52-53. MR 0271902 (42:6783)
  • [3] J. de Groot, Groups represented by homeomorphisms groups. I, Math. Ann. 138 (1959), 80-102. MR 0119193 (22:9959)
  • [4] Z. Hedrlín, Non-constant continuous transformations form any semigroup with a unity, Nieuw Arch. Wisk. 14 (1966), 230-236. MR 0209373 (35:271)
  • [5] Z. Hedrlín and J. Lambek, How comprehensive is the category of semigroups, J. Algebra 11 (1969), 195-212. MR 0237611 (38:5892)
  • [6] H. Herrlich, Wann sind alle stetigen Abbildungen in Y konstant? Math. Z. 90 (1965), 152-154. MR 0185565 (32:3029)
  • [7] -, Topologische reflexionen und coreflexionen, Lecture Notes in Math., vol. 78, Springer-Verlag, Berlin and New York, 1968. MR 0256332 (41:988)
  • [8] E. Hewitt, On two problems of Urysohn, Ann. of Math. (2) 47 (1946), 503-509. MR 0017527 (8:165g)
  • [9] J. R. Isbell, Subobjects, adequacy, completeness and categories of algebras, Rozprawy Mat. 36 (1964). MR 0163939 (29:1238)
  • [10] V. Koubek, Each concrete category has a representation by $ {T_2}$-paracompact topological spaces, Comment. Math. Univ. Carolinae 15 (1974), 655-663. MR 0354806 (50:7283)
  • [11] L. Kučera, Lectures on the theory of categories, Charles University, 1970 (preprint). (Czech)
  • [12] J. Novák, Regular space on which every continuous function is constant, Casopis Pést. Mat. Fys. 73 (1948), 58-68. MR 0028576 (10:467h)
  • [13] A. Pultr and V. Trnková, Combinatorial, algebraic and topological representations of groups, semigroups and categories, North Holland, Amsterdam, 1980; Czechoslovak Publ. House, Academia, Prague, 1980. MR 563525 (81d:18001)
  • [14] V. Trnková, Non-constant continuous mappings of metric or compact Hausdorff spaces, Comment. Math. Univ. Carolinae 13 (1972), 283-295. MR 0303486 (46:2623)
  • [15] -, All small categories are representable by continuous nonconstant mappings of bicompacta, Soviet. Math. Dokl. 17 (1976), 1403-1406.
  • [16] -, Categorial aspects are useful for topology, Proceedings of the IVth Prague Topological Symposium, Lecture Notes in Math., vol. 609, Springer-Verlag, Berlin and New York, pp. 211-215. MR 0458370 (56:16573)
  • [17] P. Vopěnka, A. Pultr and Z. Hedrlín, A rigid relation exists on any set, Comment. Math. Univ. Carolinae 6 (1965), 149-155. MR 0183647 (32:1127)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0580898-9
Article copyright: © Copyright 1980 American Mathematical Society

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