About: Finite-valued mappings preserving dimension

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
rdfs:seeAlso	https://mynsmstore.uh.edu/index.php?route=product/product&product_id=26408
Description	We say that a set-valued mapping $F:X\Rightarrow Y$ is {\em \sig{C}} provided that there exists a countable cover $\mathcal{C}$ of $X$ consisting of functionally closed sets such that for every $C\in\mathcal{C}$ and each functionally open set $U\subset Y$ one can find a functionally open set $V\subset X$ such that $\{x\in C:F(x)\cap U\neq\emptyset\}=C\cap V$. For Tychonoff spaces $X$ and $Y$ we write $X\vartriangleright Y$ provided that there exist a finite-valued \sig{C}\ mapping $F:X\Rightarrow Y$ and a finite-valued \sig{D}\ mapping $G:Y\Rightarrow X$ (for suitable $\mathcal{C}$ and $\mathcal{D}$) such that $y\in \bigcup\{F(x):x\in G(y)\}$ for every $y\in Y$. We prove that $X\vartriangleright Y$ implies $\dim X\geq\dim Y$. (Here $\dim X$ denotes the \v{C}ech-Lebesgue (covering) dimension of $X$.) As a corollary, we obtain that $\dim X=\dim Y$ whenever a perfectly normal space $Y$ is an image of a Tychonoff space $X$ under a finite-to-one open mapping. We also give an example of an open mapping $f:X\to Y$ such that $\|f^{-1}(y)\|\le 2$ for all $y\in Y$, both $X$ and $Y$ are hereditarily normal (and $Y$ is even Lindel\%22{o}f) but $\dim X\not=\dim Y$. We say that a set-valued mapping $F:X\Rightarrow Y$ is {\em \sig{C}} provided that there exists a countable cover $\mathcal{C}$ of $X$ consisting of functionally closed sets such that for every $C\in\mathcal{C}$ and each functionally open set $U\subset Y$ one can find a functionally open set $V\subset X$ such that $\{x\in C:F(x)\cap U\neq\emptyset\}=C\cap V$. For Tychonoff spaces $X$ and $Y$ we write $X\vartriangleright Y$ provided that there exist a finite-valued \sig{C}\ mapping $F:X\Rightarrow Y$ and a finite-valued \sig{D}\ mapping $G:Y\Rightarrow X$ (for suitable $\mathcal{C}$ and $\mathcal{D}$) such that $y\in \bigcup\{F(x):x\in G(y)\}$ for every $y\in Y$. We prove that $X\vartriangleright Y$ implies $\dim X\geq\dim Y$. (Here $\dim X$ denotes the \v{C}ech-Lebesgue (covering) dimension of $X$.) As a corollary, we obtain that $\dim X=\dim Y$ whenever a perfectly normal space $Y$ is an image of a Tychonoff space $X$ under a finite-to-one open mapping. We also give an example of an open mapping $f:X\to Y$ such that $\|f^{-1}(y)\|\le 2$ for all $y\in Y$, both $X$ and $Y$ are hereditarily normal (and $Y$ is even Lindel\%22{o}f) but $\dim X\not=\dim Y$. (en)
Title	Finite-valued mappings preserving dimension Finite-valued mappings preserving dimension (en)
skos:prefLabel	Finite-valued mappings preserving dimension Finite-valued mappings preserving dimension (en)
skos:notation	RIV/44555601:13440/11:43880563!RIV12-MSM-13440___
http://linked.open...avai/predkladatel	Přírodovědecká fakulta
http://linked.open...avai/riv/aktivita	I
http://linked.open...avai/riv/aktivity	I
http://linked.open...iv/cisloPeriodika	1
http://linked.open...vai/riv/dodaniDat	2012
http://linked.open...aciTvurceVysledku	Spěvák, Jan
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Univerzita Jana Evangelisty Purkyně v Ústí nad Labem / Přírodovědecká fakulta
http://linked.open...dnocenehoVysledku	199798
http://linked.open...ai/riv/idVysledku	RIV/44555601:13440/11:43880563
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	Perfectly normal space; Openmapping; Covering dimension; Lower semi-continuous mapping (en)
http://linked.open.../riv/klicoveSlovo	Covering dimension Lower semi-continuous mapping Openmapping Perfectly normal space
http://linked.open...odStatuVydavatele	US - Spojené státy americké
http://linked.open...ontrolniKodProRIV	[11867990EE4C]
http://linked.open...i/riv/nazevZdroje	Houston Journal of Mathematics
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	1 (xsd:int)
http://linked.open...UplatneniVysledku	2011
http://linked.open...v/svazekPeriodika	37
http://linked.open...iv/tvurceVysledku	Spěvák, Jan
http://linked.open...ain/vavai/riv/wos	000290812200016
issn	0362-1588
number of pages	22 (xsd:int)
http://bibframe.org/vocab/doi	10.1016/j.topol.2010.11.009
http://localhost/t...ganizacniJednotka	13440
is http://linked.open...avai/riv/vysledek of	Finite-valued mappings preserving dimension Finite-valued mappings preserving dimension

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