Binormal space: Difference between revisions

Revision as of 00:26, 25 October 2009

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

This is a variation of normal space. View other variations of normal space

Definition

Symbol-free definition

A topological space is termed binormal if it satisfies the following two conditions:

Its product with the unit interval (with the product topology) is a normal space.
it is both a normal space and a countably paracompact space.

Relation with other properties

Stronger properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Compact Hausdorff space
Paracompact Hausdorff space

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Normal space				\|FULL LIST, MORE INFO

References

Textbook references

Algebraic Topology by Edwin H. Spanier^{More info}, Page 56, Exercise B-2 (definition introduced in exercise)

@@ Line 1: / Line 1: @@
 {{topospace property}}
-{{variationof|normality}}
+{{variation of|normal space}}
 ==Definition==
 ===Symbol-free definition===
-A [[topological space]] is termed '''binormal''' if it is a [[normal space]] and its direct product with the [[unit interval]] is also a normal space.
+A [[topological space]] is termed '''binormal''' if it satisfies the following two conditions:
+# Its product with the [[unit interval]] (with the [[product topology]]) is a [[normal space]].
+# it is both a [[defining ingredient::normal space]] and a [[defining ingredient::countably paracompact space]].
 ==Relation with other properties==
 ===Stronger properties===
-* [[Compact Hausdorff space]]
+{| class="wikitable" border="1"
-* [[Paracompact Hausdorff space]]
+! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
+|-
+| [[Weaker than::Compact Hausdorff space]] || || || ||
+|-
+| [[Weaker than::Paracompact Hausdorff space]] || || || ||
+|}
 ===Weaker properties===
-* [[Normal space]]
+{| class="wikitable" border="1"
+! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
+|-
+| [[Stronger than::Normal space]] || || || || {{intermediate notions short|normal space|binormal space}}
+|}
 ==References==
 ===Textbook references===
 * {{booklink|Spanier}}, Page 56, Exercise B-2 (definition introduced in exercise)