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A continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f0419301.png" /> with the following property: For any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f0419302.png" /> and for any finite family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f0419303.png" /> of open subsets of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f0419304.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f0419305.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f0419306.png" /> is open. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f0419307.png" /> denotes the [[Small image|small image]] of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f0419308.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f0419309.png" />. Any fully-closed mapping is closed. The inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193010.png" /> is valid for any fully-closed mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193011.png" /> of a normal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193012.png" />. For this reason, fully-closed mappings can be employed to isolate fairly wide classes of compacta with non-coinciding dimensions dim and ind. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193013.png" /> irrespective of the multiplicity of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193014.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193015.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193016.png" /> be a fully-closed mapping and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193017.png" /> be the decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193018.png" /> the elements of which are all pre-images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193019.png" /> of the points, and all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193020.png" />. Then, for a regular space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193021.png" />, the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193023.png" /> with respect to the decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041930/f04193024.png" /> is also regular; this property is characteristic of fully-closed mappings in the class of closed mappings.
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A continuous mapping  $  f:  X \rightarrow Y $
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with the following property: For any point $  y \in Y $
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and for any finite family $  \{ O _ {1} \dots O _ {s} \} $
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of open subsets of the space $  X $
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such that $  f ^ { - 1 } ( y) = {\cup _ {i=1}  ^ {s} } O _ {i} $,  
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the set $  \{ y \} \cup ( {\cup _ {i=1}  ^ {s} } f ^ { \sharp } O _ {i} ) $
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is open. Here f ^ { \sharp } O _ {i} $
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denotes the [[small image]] of the set $  O _ {i} $
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under the mapping f $.  
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Any fully-closed mapping is closed. The inequality $  \mathop{\rm dim}  X \leq  \max \{  \mathop{\rm dim}  Y ,  \mathop{\rm dim}  f \} $
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is valid for any fully-closed mapping $  f: X \rightarrow Y $
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of a normal space $  X $.  
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For this reason, fully-closed mappings can be employed to isolate fairly wide classes of compacta with non-coinciding dimensions dim and ind. Moreover, $  \mathop{\rm dim}  Y \leq  \mathop{\rm dim}  X+ 1 $
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irrespective of the multiplicity of the mapping f $.  
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Let $  y \in Y $,  
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let $  f: X \rightarrow Y $
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be a fully-closed mapping and let $  R( f, y) $
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be the decomposition of $  X $
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the elements of which are all pre-images f ^ { - 1 } ( y  ^  \prime  ) $
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of the points, and all points of f ^ { - 1 } ( y) $.  
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Then, for a regular space $  X $,  
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the quotient space $  X _ {f}  ^ {y} $
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of $  X $
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with respect to the decomposition $  R( f, y) $
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is also regular; this property is characteristic of fully-closed mappings in the class of closed mappings.

Latest revision as of 20:01, 22 February 2021


A continuous mapping $ f: X \rightarrow Y $ with the following property: For any point $ y \in Y $ and for any finite family $ \{ O _ {1} \dots O _ {s} \} $ of open subsets of the space $ X $ such that $ f ^ { - 1 } ( y) = {\cup _ {i=1} ^ {s} } O _ {i} $, the set $ \{ y \} \cup ( {\cup _ {i=1} ^ {s} } f ^ { \sharp } O _ {i} ) $ is open. Here $ f ^ { \sharp } O _ {i} $ denotes the small image of the set $ O _ {i} $ under the mapping $ f $. Any fully-closed mapping is closed. The inequality $ \mathop{\rm dim} X \leq \max \{ \mathop{\rm dim} Y , \mathop{\rm dim} f \} $ is valid for any fully-closed mapping $ f: X \rightarrow Y $ of a normal space $ X $. For this reason, fully-closed mappings can be employed to isolate fairly wide classes of compacta with non-coinciding dimensions dim and ind. Moreover, $ \mathop{\rm dim} Y \leq \mathop{\rm dim} X+ 1 $ irrespective of the multiplicity of the mapping $ f $. Let $ y \in Y $, let $ f: X \rightarrow Y $ be a fully-closed mapping and let $ R( f, y) $ be the decomposition of $ X $ the elements of which are all pre-images $ f ^ { - 1 } ( y ^ \prime ) $ of the points, and all points of $ f ^ { - 1 } ( y) $. Then, for a regular space $ X $, the quotient space $ X _ {f} ^ {y} $ of $ X $ with respect to the decomposition $ R( f, y) $ is also regular; this property is characteristic of fully-closed mappings in the class of closed mappings.

How to Cite This Entry:
Fully-closed mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fully-closed_mapping&oldid=14852
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article