The basis criterion (topology)
From Maths
(Redirected from Basis criterion (topology))
Contents
[<hidetoc>]Statement
Let (X,J) be a topological space and let B∈P(P(X)) be a topological basis for (X,J). Then[1]:
- ∀U∈P(X)[U∈J⟺∀p∈U∃B∈B[p∈B⊆U]⏟basis criterion][Note 1]
If a subset U of X satisfies[Note 2] ∀p∈U∃B∈B[p∈B⊆U] we say it satisfies the basis criterion with respect to B[1]
Proof
Grade: A
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
This proof has been marked as an page requiring an easy proof
The message provided is:
Can't be bothered to do now, it's a minor restatement of the second property of a basis.
This proof has been marked as an page requiring an easy proof
Notes
- <cite_references_link_accessibility_label> ↑ Note that when we write p∈B⊆U we actually mean p∈B∧B⊆U. This is a very slight abuse of notation and the meaning of what is written should be obvious to all without this note
- <cite_references_link_accessibility_label> ↑ This means "if a U∈P(X) satisfies...
References
- ↑ <cite_references_link_many_accessibility_label> 1.0 1.1 Introduction to Topological Manifolds - John M. Lee
| |||||||||||||||||||||||
