Functions characterized by images of sets

Ciesielski, Krzysztof, Dikrajan, Dikran, and Watson, Stephen. "Functions characterized by images of sets." Colloquium Mathematicae 77.2 (1998): 211-232. <http://eudml.org/doc/210585>.

@article{Ciesielski1998,
abstract = {For non-empty topological spaces X and Y and arbitrary families $\mathcal \{A\}$ ⊆ $\mathcal \{P\}(X)$ and $\mathcal \{B\} ⊆ \mathcal \{P\}(Y)$ we put $\mathcal \{C\}_\{\mathcal \{A\},\mathcal \{B\}\}$=f ∈ $Y^X$ : (∀ A ∈ $\mathcal \{A\}$)(f[A] ∈ $\mathcal \{B\})$. We examine which classes of functions $\mathcal \{F\}$ ⊆ $Y^X$ can be represented as $\mathcal \{C\}_\{\mathcal \{A\},\mathcal \{B\}\}$. We are mainly interested in the case when $\mathcal \{F\}=\mathcal \{C\}(X,Y)$ is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class $\mathcal \{F\}=\mathcal \{C\}$(X,ℝ) is not equal to $\mathcal \{C\}_\{\mathcal \{A\},\mathcal \{B\}\}$ for any $\mathcal \{A\}$ ⊆ $\mathcal \{P\}(X)$ and $\mathcal \{B\}$ ⊆ $\mathcal \{P\}$(ℝ). Thus, $\mathcal \{C\}$(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 $\mathcal \{C\}_\{\mathcal \{A\},\mathcal \{B\}\}$: upper (lower) semicontinuous functions, derivatives, approximately continuous functions, Baire class 1 functions, Borel functions, and measurable functions.},
author = {Ciesielski, Krzysztof, Dikrajan, Dikran, Watson, Stephen},
journal = {Colloquium Mathematicae},
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; classes of functions; continuous functions},
language = {eng},
number = {2},
pages = {211-232},
title = {Functions characterized by images of sets},
url = {http://eudml.org/doc/210585},
volume = {77},
year = {1998},
}

TY - JOUR
AU - Ciesielski, Krzysztof
AU - Dikrajan, Dikran
AU - Watson, Stephen
TI - Functions characterized by images of sets
JO - Colloquium Mathematicae
PY - 1998
VL - 77
IS - 2
SP - 211
EP - 232
AB - For non-empty topological spaces X and Y and arbitrary families $\mathcal {A}$ ⊆ $\mathcal {P}(X)$ and $\mathcal {B} ⊆ \mathcal {P}(Y)$ we put $\mathcal {C}_{\mathcal {A},\mathcal {B}}$=f ∈ $Y^X$ : (∀ A ∈ $\mathcal {A}$)(f[A] ∈ $\mathcal {B})$. We examine which classes of functions $\mathcal {F}$ ⊆ $Y^X$ can be represented as $\mathcal {C}_{\mathcal {A},\mathcal {B}}$. We are mainly interested in the case when $\mathcal {F}=\mathcal {C}(X,Y)$ is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class $\mathcal {F}=\mathcal {C}$(X,ℝ) is not equal to $\mathcal {C}_{\mathcal {A},\mathcal {B}}$ for any $\mathcal {A}$ ⊆ $\mathcal {P}(X)$ and $\mathcal {B}$ ⊆ $\mathcal {P}$(ℝ). Thus, $\mathcal {C}$(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 $\mathcal {C}_{\mathcal {A},\mathcal {B}}$: upper (lower) semicontinuous functions, derivatives, approximately continuous functions, Baire class 1 functions, Borel functions, and measurable functions.
LA - eng
KW - 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; classes of functions; continuous functions
UR - http://eudml.org/doc/210585
ER -