# Functions characterized by images of sets

Krzysztof Ciesielski; Dikran Dikrajan; Stephen Watson

Colloquium Mathematicae (1998)

- Volume: 77, Issue: 2, page 211-232
- ISSN: 0010-1354

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topCiesielski, 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 -

## References

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- [3] H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1967), 241-249. Zbl0158.41503
- [4] R. Engelking, General Topology, PWN, Warszawa, 1977.
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- [6] V. Kannan and M. Rajagopalan, Construction and application of rigid spaces I, Adv. Math. 29 (1978), 1139-1172. Zbl0424.54028
- [7] W. Kulpa, Rigid graphs of maps, Ann. Math. Sil. 2 (14) (1986), 92-95. Zbl0593.54013
- [8] D. J. Velleman, Characterizing continuity, Amer. Math. Monthly 104 (1997), 318-322. Zbl0871.26002

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