Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
compactness; connectedness; preserving compactness; preserving connectedness
compactness; connectedness; preserving compactness; preserving connectedness
Summary:
We call a function $f\colon X\to Y\,$ P-preserving if, for every subspace $A \subset X$ with property P, its image $f(A)$ also has property P. Of course, all continuous maps are both compactness- and connectedness-preserving and the natural question about when the converse of this holds, i.e. under what conditions such a map is continuous, has a long history. Our main result is that any nontrivial product function, i.e. one having at least two nonconstant factors, that has connected domain, $T_1$ range, and is connectedness-preserving must actually be continuous. The analogous statement badly fails if we replace in it the occurrences of "connected" by "compact". We also present, however, several interesting results and examples concerning maps that are compactness-preserving and/or continuum-preserving.
References:
[1] Gerlits J., Juhász I., Soukup L., Szentmiklóssy Z.: Characterizing continuity by preserving compactness and connectedness. Topology Appl. 138 (2004), no. 1–3, 21–44. DOI 10.1016/j.topol.2003.07.005 | MR 1906831
[2] Hart J. E., Kunen K.: One-dimensional locally connected $S$-spaces. Topology Appl. 156 (2009), no. 3, 601–609. DOI 10.1016/j.topol.2008.08.009 | MR 2492307
[3] McMillan E. R.: On continuity conditions for functions. Pacific J. Math. 32 (1970), 479–494. DOI 10.2140/pjm.1970.32.479 | MR 0257986
[4] van Mill J.: A locally connected continuum without convergent sequences. Topology Appl. 126 (2002), no. 1–2, 273–280. DOI 10.1016/S0166-8641(02)00088-3 | MR 1934264
[5] Nadler S. B.: Continuum Theory. An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, New York, 1992. MR 1192552 | Zbl 0819.54015
[6] White D. J.: Functions preserving compactness and connectedness. J. London Math. Soc. 3 (1971), 767–768. DOI 10.1112/jlms/s2-3.4.767 | MR 0290345
Search
Partner of
