Binormal space: Difference between revisions
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{{topospace property}} | {{topospace property}} | ||
{{variation of|normal space}} | |||
==Definition== | ==Definition== | ||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[topological space]] is termed '''binormal''' if it is a [[normal space]] and | A [[topological space]] is termed '''binormal''' if its product with the [[unit interval]] (with the [[product topology]]) is a [[normal space]]. | ||
Equivalently (??), a topological space is termed binormal if it is both a [[defining ingredient::normal space]] and a [[defining ingredient::countably paracompact space]]. | |||
==Relation with other properties== | ==Relation with other properties== | ||
===Stronger properties=== | |||
{| class="wikitable" border="1" | |||
! 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=== | ===Weaker properties=== | ||
{| 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) | |||
Latest revision as of 15:08, 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 its product with the unit interval (with the product topology) is a normal space.
Equivalently (??), a topological space is termed binormal if 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. SpanierMore info, Page 56, Exercise B-2 (definition introduced in exercise)