Normally-imbedded subspace
2020 Mathematics Subject Classification: Primary: 54-XX [MSN][ZBL]
A subspace 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 |
Comments
A finally-compact space is the same as a Lindelöf space.
Normally-imbedded subspace. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normally-imbedded_subspace&oldid=34638