Difference between revisions of "Totally-normal space"
From Encyclopedia of Mathematics
(Importing text file) |
(TeX) |
||
| Line 1: | Line 1: | ||
| − | A [[Topological space|topological space]] in which for any two subsets | + | {{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==== | ====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 | + | 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==== | ====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> | <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> | ||
Latest revision as of 16:35, 1 May 2014
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) |
How to Cite This Entry:
Totally-normal space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Totally-normal_space&oldid=16218
Totally-normal space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Totally-normal_space&oldid=16218
This article was adapted from an original article by V.I. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article