# Normally-imbedded subspace

From Encyclopedia of Mathematics

A subspace of a space such that for every neighbourhood of it in there is a set that is the union of a countable family of sets closed in and with . If is normally imbedded in and is normally imbedded in , then is normally imbedded in . 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=11245

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