Difference between revisions of "Normally-imbedded subspace"
From Encyclopedia of Mathematics
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| − | A subspace | + | 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> | + | <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> | ||
| + | ====Comments==== | ||
| + | A finally-compact space is the same as a [[Lindelöf space]]. | ||
| − | + | {{TEX|done}} | |
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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
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