# Normally-imbedded subspace

2010 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.

#### References

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