- Tytuł:
- Functions characterized by images of sets
- Autorzy:
-
Ciesielski, Krzysztof
Dikrajan, Dikran
Watson, Stephen - Powiązania:
- https://bibliotekanauki.pl/articles/966079.pdf
- Data publikacji:
- 1998
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Tematy:
-
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 - Opis:
- 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.
- Źródło:
-
Colloquium Mathematicum; 1998, 77, 2; 211-232
0010-1354 - Pojawia się w:
- Colloquium Mathematicum
- Dostawca treści:
- Biblioteka Nauki
Artykuł