Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-14T05:12:09.312Z Has data issue: false hasContentIssue false
  • English
  • Français

Spaces on which every Continuous Map into a given Space is Constant

Published online by Cambridge University Press:  20 November 2018

S. Iliadis  and
V. Tzannes
S. Iliadis
Affiliation:
University of Patras, Fatras, Greece
V. Tzannes
Affiliation:
University of Patras, Fatras, Greece
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is concerned with topological spaces whose continuous maps into a given space R are constant, as well as with spaces having this property locally. We call these spaces R-monolithic and locally R-monolithic, respectively.

Spaces with such properties have been considered in [1], [5]-[7], [10], [11], [22], [28], [31], where with the exception of [10], the given space R is the set of real-numbers with the usual topology. Obviously, for a countable space, connectedness is equivalent to the property that every continuous real-valued map is constant. Countable connected (locally connected) spaces have been constructed in papers [2]-[4], [8], [9], [11]-[21], [23]-[26], [30].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 6 , 01 December 1986 , pp. 1281 - 1298
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Armentrout, S., A Moore space on which every real-valued continuous function is constant, Proc. Amer. Math. Soc. 12 (1961), 106–109.Google Scholar
2. Baggs, I., A connected Hausdorff space which is not contained in a maximal connected space, Pacific J. of Math. 57 (1974).Google Scholar
3. Bing, R. H., A connected countable Hausdorff space, Proc. Amer. Math. Soc. 4 (1953), 474.Google Scholar
4. Brown, M., A countable connected Hausdorff space, Bull. Amer. Math. Soc. 59 (1953), 367.Google Scholar
5. van Douwen, E. K., A regular space on which every continuous real-valued function is constant, Nieuw Archief voor Wiskunde 20 (1972), 143–145.Google Scholar
6. van Est, W. T. and Freudenthal, U. H., Trennung durch stetige Funktionen in Topologischen Räurnen, Indagationes Math. 13 (1951), 359–368.Google Scholar
7. Gantner, T. E., A regular space on which every continuous real-valued function is constant, Amer. Math. Monthly 75 (1971), 52–53.Google Scholar
8. Colomb, S. W., A connected topology for the integers, Amer. Math. Monthly 66 (1959), 663–665.Google Scholar
9. Gustin, W., Countable connected spaces, Bull. Amer. Math. Soc. 52 (1946), 101–106.Google Scholar
10. Herrlich, H., Warm sind aile stetigen Abbildungen in Y konstant?, Math. Zeitschr. 90 (1965), 152–154.Google Scholar
11. Hewitt, E., On two problems of Urysohn, Annals of Mathematics 47 (1946).Google Scholar
12. Jones, F. B. and Stone, A. H., Countable locally connected Urysohn spaces, Colloq. Math. 22 (1971), 239–244.Google Scholar
13. Kannan, V. and Rajagopalan, M., On countable locally connected spaces, Colloq. Math. 29 (1974), 93–100.Google Scholar
14. Kirch, A. M., A countable connected, locally connected Hausdorff space, Amer. Math. Monthly 76 (1969), 169–171.Google Scholar
15. Larmore, L., A connected countable Hausdorff space, Sα for every countable ordinal α, Bol. Soc. Mat. Mexicana 17 (1972), 14–17.Google Scholar
16. Martin, J., A countable Hausdorff space with a dispersion point, Duke Math. J. 33 (1966), 165–167.Google Scholar
17. Michaelides, G. J., On countable connected Hausdorff spaces, Studies and Essays, Mathematics Research Center, National Taiwan University Taipei, Taiwan, China, (1970), 183–189.Google Scholar
18. Miller, G. G., Countable connected spaces, Proc. Amer. Math. Soc. 26 (1970), 355–360.Google Scholar
19. Miller, G. G., A countable Urysohn space with an explosion point, Notices Amer. Math. Soc. 13 (1966), 589.Google Scholar
20. Miller, G. G., A countable locally connected quasimetric space, Notices Amer. Math. Soc. 14 (1967), 720.Google Scholar
21. Miller, G. and Pearson, B. J., On the connectification of a space by a countable point set, J. Austral. Math. Soc. 13 (1972), 67–70.Google Scholar
22. Novák, J., Regular space on which every continuous function is constant, (English summary) Časopis est Mat. Fys. 73 (1948), 58–68.Google Scholar
23. Ritter, G. X., A connected, locally connected, countable Hausdorff space, Amer. Math. Monthly 83 (1976), 185–186.Google Scholar
24. Ritter, G. X., A connected, locally connected, countable Urysohn space, Gen. Topology Appl. 7 (1977), 65–70.Google Scholar
25. Roy, P., A countable connected Urysohn space with a dispersion point, Duke Math. J. 33 (1966), 331–334.Google Scholar
26. Stone, A. H., A countable connected, locally connected Hausdorff space, Notices Amer. Math. Soc. 16 (1969), 442.Google Scholar
27. Thomas, J., A regular space not completely regular, Amer. Math. Monthly 76 (1969), 181–182.Google Scholar
28. Urysohn, P., Über die Mächtigkeit der zusammenhängenden Mengen, Math. Ann. 94 (1925), 262–295.Google Scholar
29. Vickery, C. W., Axioms for Moore spaces and metric spaces, Bull. Amer. Math. Soc. 46 (1940), 560–564.Google Scholar
30. Vought, E. J., A countable connected Urysohn space with a dispersion point that is regular almost everywhere, Colloq. Math. 28 (1973), 205–209.Google Scholar
31. Younglove, J. N., A locally connected, complete Moore space on which every real-valued continuous function is constant, Proc. Amer. Math. Soc. 20 (1969), 527–530.Google Scholar