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Difference between revisions of "Local homeomorphism"

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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l0601401.png" /> between topological spaces such that for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l0601402.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l0601403.png" /> that maps homeomorphically into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l0601404.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l0601405.png" /> (cf. [[Homeomorphism|Homeomorphism]]). Sometimes in the definition of a local homeomorphism the requirement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l0601406.png" /> is included and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l0601407.png" /> is also assumed to be open (cf. [[Open mapping|Open mapping]]). Examples of local homeomorphisms are: a continuously-differentiable mapping with non-zero Jacobian on an open subset of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l0601408.png" />-dimensional Euclidean space into the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l0601409.png" />-dimensional Euclidean space; a covering mapping, in particular the natural mapping of a topological group onto its quotient space with respect to a discrete subgroup. If the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l06014010.png" /> of a Čech-complete space, in particular a locally compact Hausdorff space, onto a Tikhonov space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l06014011.png" /> is open and countable-to-one, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l06014012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l06014013.png" />, then on some open everywhere-dense set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l06014014.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060140/l06014015.png" /> is a local homeomorphism.
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A mapping $f : X \rightarrow Y$ between topological spaces such that for every point $x \in X$ there is a neighbourhood $\mathcal{O}_x$ that maps homeomorphically into $Y$ under $f$ (cf. [[Homeomorphism]]). Sometimes in the definition of a local homeomorphism the requirement $f(X) = Y$ is included and $f$ is also assumed to be an [[open mapping]]. Examples of local homeomorphisms are: a continuously-differentiable mapping with non-zero Jacobian on an open subset of an $n$-dimensional Euclidean space into the $n$-dimensional Euclidean space; a covering mapping, in particular the natural mapping of a topological group onto its quotient space with respect to a discrete subgroup. If the mapping $f : X \rightarrow Y$ of a [[Čech-complete space]], in particular a locally compact Hausdorff space, onto a [[Tikhonov space]] $Y$ is open and countable-to-one, that is, $|f^{-1}(y)| \le \aleph_0$, $y \in Y$, then on some open everywhere-dense set in $X$ the mapping $f$ is a local homeomorphism.
  
  
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====References====
 
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR>
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Latest revision as of 19:37, 1 November 2016

A mapping $f : X \rightarrow Y$ between topological spaces such that for every point $x \in X$ there is a neighbourhood $\mathcal{O}_x$ that maps homeomorphically into $Y$ under $f$ (cf. Homeomorphism). Sometimes in the definition of a local homeomorphism the requirement $f(X) = Y$ is included and $f$ is also assumed to be an open mapping. Examples of local homeomorphisms are: a continuously-differentiable mapping with non-zero Jacobian on an open subset of an $n$-dimensional Euclidean space into the $n$-dimensional Euclidean space; a covering mapping, in particular the natural mapping of a topological group onto its quotient space with respect to a discrete subgroup. If the mapping $f : X \rightarrow Y$ of a Čech-complete space, in particular a locally compact Hausdorff space, onto a Tikhonov space $Y$ is open and countable-to-one, that is, $|f^{-1}(y)| \le \aleph_0$, $y \in Y$, then on some open everywhere-dense set in $X$ the mapping $f$ is a local homeomorphism.


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References

[a1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
How to Cite This Entry:
Local homeomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_homeomorphism&oldid=39586
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article