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Functions characterized by images of sets

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Ciesielski K., Dikrajan D., Watson S.

Colloquium Mathematicum

1998

tom 77

nr 2

211-232

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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Ciesielski K.

1 Dikrajan D.

1 Watson S.

1 1998

Wyniki wyszukiwania

Functions characterized by images of sets