View source for Normally-imbedded subspace

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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====
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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) {{ZBL|0043.16502}}</TD></TR>
</table>

====Comments====
A finally-compact space is the same as a [[Lindelöf space]].

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