On maps preserving connectedness and/or compactness

István Juhász; Jan van Mill

Commentationes Mathematicae Universitatis Carolinae (2018)

  • Volume: 59, Issue: 4, page 513-521
  • ISSN: 0010-2628

Abstract

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We call a function f : X → Y P-preserving if, for every subspace A ⊂ 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.

How to cite

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Juhász, István, and van Mill, Jan. "On maps preserving connectedness and/or compactness." Commentationes Mathematicae Universitatis Carolinae 59.4 (2018): 513-521. <http://eudml.org/doc/294683>.

@article{Juhász2018,
abstract = {We call a function $f\colon X\rightarrow Y\,$ 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.},
author = {Juhász, István, van Mill, Jan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {compactness; connectedness; preserving compactness; preserving connectedness},
language = {eng},
number = {4},
pages = {513-521},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On maps preserving connectedness and/or compactness},
url = {http://eudml.org/doc/294683},
volume = {59},
year = {2018},
}

TY - JOUR
AU - Juhász, István
AU - van Mill, Jan
TI - On maps preserving connectedness and/or compactness
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2018
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 59
IS - 4
SP - 513
EP - 521
AB - We call a function $f\colon X\rightarrow Y\,$ 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.
LA - eng
KW - compactness; connectedness; preserving compactness; preserving connectedness
UR - http://eudml.org/doc/294683
ER -

References

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  4. van Mill J., 10.1016/S0166-8641(02)00088-3, Topology Appl. 126 (2002), no. 1–2, 273–280. MR1934264DOI10.1016/S0166-8641(02)00088-3
  5. Nadler S. B., Continuum Theory, An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, New York, 1992. Zbl0819.54015MR1192552
  6. White D. J., 10.1112/jlms/s2-3.4.767, J. London Math. Soc. 3 (1971), 767–768. MR0290345DOI10.1112/jlms/s2-3.4.767

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