Totally-normal space

A topological space in which for any two subsets $A$, $B$ satisfying the conditions $[A]\cap B=\emptyset$, $A\cap[B]=\emptyset$ there are disjoint neighbourhoods; here, $[A]$ and $[B]$ are the closures of the sets $A$ and $B$, while $\emptyset$ is the empty set. Totally-normal spaces and only such spaces are hereditarily normal. Perfectly-normal spaces (cf. Perfectly-normal space) are totally normal, but the converse is not true. Normal spaces (cf. Normal space) which are not totally normal also exist.

Comments

In the West, these spaces are called completely normal. A totally-normal space is a normal space each of whose open sets is the union of a locally finite family of open $F_\sigma$'s, [a1]. Thus, these spaces generalize perfectly-normal spaces. Much of what can be done, in dimension and homology theory, for perfectly-normal spaces generalizes to totally-normal spaces in this sense.

References

[a1]	C.H. Dowker, "Inductive dimension of completely normal spaces" Quart. J. Math. (Oxford) , 4 (1952) pp. 267–281
[a2]	R. Engelking, "General topology" , Heldermann (1989)