Functions characterized by images of sets
Ciesielski, Krzysztof ; Dikrajan, Dikran ; Watson, Stephen
Colloquium Mathematicae, Tome 78 (1998), p. 211-232 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

For non-empty topological spaces X and Y and arbitrary families 𝒜 ⊆ 𝒫(X) and ℬ⊆𝒫(Y) we put 𝒞𝒜,ℬ=f ∈ YX : (∀ A ∈ 𝒜)(f[A] ∈ ℬ). We examine which classes of functions ℱ ⊆ YX can be represented as 𝒞𝒜,ℬ. We are mainly interested in the case when ℱ=𝒞(X,Y) is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class ℱ=𝒞(X,ℝ) is not equal to 𝒞𝒜,ℬ for any 𝒜 ⊆ 𝒫(X) and ℬ ⊆ 𝒫(ℝ). Thus, 𝒞(X,ℝ) cannot be characterized by images of sets. We also show that none of the following classes of real functions can be represented as 𝒞𝒜,ℬ: upper (lower) semicontinuous functions, derivatives, approximately continuous functions, Baire class 1 functions, Borel functions, and measurable functions.

Publié le : 1998-01-01
EUDML-ID : urn:eudml:doc:210585 
@article{bwmeta1.element.bwnjournal-article-cmv77z2p211bwm,
     author = {Krzysztof Ciesielski and Dikran Dikrajan and Stephen Watson},
     title = {Functions characterized by images of sets},
     journal = {Colloquium Mathematicae},
     volume = {78},
     year = {1998},
     pages = {211-232},
     zbl = {0909.54008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv77z2p211bwm}
}

Ciesielski, Krzysztof; Dikrajan, Dikran; Watson, Stephen. Functions characterized by images of sets. Colloquium Mathematicae, Tome 78 (1998) pp. 211-232. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv77z2p211bwm/

[000] [1] J. J. Charatonik, On chaotic curves, Colloq. Math. 41 (1979), 219-227. | Zbl 0447.54047

[001] [2] K. Ciesielski, Topologizing different classes of real functions, Canad. J. Math. 46 (1994), 1188-1207. | Zbl 0828.26011

[002] [3] H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1967), 241-249. | Zbl 0158.41503

[003] [4] R. Engelking, General Topology, PWN, Warszawa, 1977.

[004] [5] H. Herrlich, Wann sind alle stetigen Abbildungen in Y konstant?, Math. Z. 90 (1965), 152-154. | Zbl 0131.20402

[005] [6] V. Kannan and M. Rajagopalan, Construction and application of rigid spaces I, Adv. Math. 29 (1978), 1139-1172. | Zbl 0424.54028

[006] [7] W. Kulpa, Rigid graphs of maps, Ann. Math. Sil. 2 (14) (1986), 92-95. | Zbl 0593.54013

[007] [8] D. J. Velleman, Characterizing continuity, Amer. Math. Monthly 104 (1997), 318-322. | Zbl 0871.26002