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Difference between revisions of "Normally-imbedded subspace"

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A subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067750/n0677501.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067750/n0677502.png" /> such that for every neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067750/n0677503.png" /> of it in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067750/n0677504.png" /> there is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067750/n0677505.png" /> that is the union of a countable family of sets closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067750/n0677506.png" /> and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067750/n0677507.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067750/n0677508.png" /> is normally imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067750/n0677509.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067750/n06775010.png" /> is normally imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067750/n06775011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067750/n06775012.png" /> is normally imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067750/n06775013.png" />. A normally-imbedded subspace of a [[Normal space|normal space]] is itself normal in the induced topology, which explains the name. Final compactness of a space is equivalent to its being normally imbedded in some (and hence in any) [[Compactification|compactification]] of this space. Quite generally, a normally-imbedded subspace of a finally-compact space is itself finally compact.
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A subspace $A$ of a space $X$ such that for every neighbourhood $U$ of $A$ in $X$ there is a set $H$ that is the union of a countable family of sets closed in $X$ and with $A \subset H \subset U$. If $A$ is normally imbedded in $X$ and $X$ is normally imbedded in $Y$, then $A$ is normally imbedded in $Y$. A normally-imbedded subspace of a [[normal space]] is itself normal in the [[induced topology]], which explains the name. Final compactness of a space is equivalent to its being normally imbedded in some (and hence in any) [[compactification]] of this space. Quite generally, a normally-imbedded subspace of a finally-compact space is itself finally compact.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.M. Smirnov,  "On normally-imbedded sets of normal spaces"  ''Mat. Sb.'' , '''29'''  (1951)  pp. 173–176  (In Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.M. Smirnov,  "On normally-imbedded sets of normal spaces"  ''Mat. Sb.'' , '''29'''  (1951)  pp. 173–176  (In Russian)</TD></TR>
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</table>
  
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====Comments====
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A finally-compact space is the same as a [[Lindelöf space]].
  
 
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{{TEX|done}}
====Comments====
 
A finally-compact space is the same as a [[Lindelöf space|Lindelöf space]].
 

Revision as of 15:24, 20 November 2014

A subspace $A$ of a space $X$ such that for every neighbourhood $U$ of $A$ in $X$ there is a set $H$ that is the union of a countable family of sets closed in $X$ and with $A \subset H \subset U$. If $A$ is normally imbedded in $X$ and $X$ is normally imbedded in $Y$, then $A$ is normally imbedded in $Y$. A normally-imbedded subspace of a normal space is itself normal in the induced topology, which explains the name. Final compactness of a space is equivalent to its being normally imbedded in some (and hence in any) compactification of this space. Quite generally, a normally-imbedded subspace of a finally-compact space is itself finally compact.

References

[1] Yu.M. Smirnov, "On normally-imbedded sets of normal spaces" Mat. Sb. , 29 (1951) pp. 173–176 (In Russian)

Comments

A finally-compact space is the same as a Lindelöf space.

How to Cite This Entry:
Normally-imbedded subspace. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normally-imbedded_subspace&oldid=34634
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article