Functions characterized by images of sets

Krzysztof Ciesielski¹ ; Dikran Dikrajan¹ ; Stephen Watson¹

¹

Colloquium Mathematicum, Tome 77 (1998) no. 2, pp. 211-232.

Résumé

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.

Détail

DOI : 10.4064/cm-77-2-211-232

Keywords: continuous function, strongly rigid family of spaces, upper or lower semicontinuous function, Tikhonov space, derivative, Borel function, Baire class 1 function, Cook continuum, measurable function, approximately continuous function, functionally Hausdorff space