Normally-imbedded subspace

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2020 Mathematics Subject Classification: Primary: 54-XX [MSN][ZBL]

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.


[1] Yu.M. Smirnov, "On normally-imbedded sets of normal spaces" Mat. Sb. , 29 (1951) pp. 173–176 (In Russian) Zbl 0043.16502


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

How to Cite This Entry:
Normally-imbedded subspace. Encyclopedia of Mathematics. URL:
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