View source for Totally-normal space

{{TEX|done}}
A [[Topological space|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|Perfectly-normal space]]) are totally normal, but the converse is not true. Normal spaces (cf. [[Normal space|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, [[#References|[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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.H. Dowker,   "Inductive dimension of completely normal spaces"  ''Quart. J. Math. (Oxford)'' , '''4'''  (1952)  pp. 267–281</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Engelking,   "General topology" , Heldermann  (1989)</TD></TR></table>