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1998 | 77 | 2 | 211-232
Tytuł artykułu

Functions characterized by images of sets

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Języki publikacji
EN
Abstrakty
EN
For non-empty topological spaces X and Y and arbitrary families $\cal A$ ⊆ $\cal P(X)$ and $\cal B ⊆ \cal P(Y)$ we put $\cal C_{\cal A,\cal B}$={f ∈ $Y^X$ : (∀ A ∈ $\cal A$)(f[A] ∈ $\cal B)$}. We examine which classes of functions $\cal F$ ⊆ $Y^X$ can be represented as $\cal C_{\cal A,\cal B}$. We are mainly interested in the case when $\cal F=\cal C(X,Y)$ is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class $\cal F=\cal C$(X,ℝ) is not equal to $\cal C_{\cal A,\cal B}$ for any $\cal A$ ⊆ $\cal P(X)$ and $\cal B$ ⊆ $\cal P$(ℝ). Thus, $\cal C$(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 $\cal C_{\cal A,\cal B}$: upper (lower) semicontinuous functions, derivatives, approximately continuous functions, Baire class 1 functions, Borel functions, and measurable functions.
Rocznik
Tom
77
Numer
2
Strony
211-232
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-12-03
Twórcy
  • Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310, U.S.A.
  • Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy
  • Department of Mathematics and Statistics York, University Toronto, Ontario Canada
Bibliografia
  • [1] J. J. Charatonik, On chaotic curves, Colloq. Math. 41 (1979), 219-227.
  • [2] K. Ciesielski, Topologizing different classes of real functions, Canad. J. Math. 46 (1994), 1188-1207.
  • [3] H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1967), 241-249.
  • [4] R. Engelking, General Topology, PWN, Warszawa, 1977.
  • [5] H. Herrlich, Wann sind alle stetigen Abbildungen in Y konstant?, Math. Z. 90 (1965), 152-154.
  • [6] V. Kannan and M. Rajagopalan, Construction and application of rigid spaces I, Adv. Math. 29 (1978), 1139-1172.
  • [7] W. Kulpa, Rigid graphs of maps, Ann. Math. Sil. 2 (14) (1986), 92-95.
  • [8] D. J. Velleman, Characterizing continuity, Amer. Math. Monthly 104 (1997), 318-322.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-cmv77z2p211bwm
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