Notes:Basis for a topology/McCarty
Contents
Overview
After finding out that base/subbase are terms (and the book I saw them in wasn't the odd one out) I've decided to note what the book says here.
Things make a lot more sense now.
Statements
Definition: Basis / Base
Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, let [ilmath]\mathcal{B}\subseteq\mathcal{J} [/ilmath]. We say [ilmath]\mathcal{B} [/ilmath] is a base or basis for [ilmath]\mathcal{J} [/ilmath] if[1]:
- [ilmath]\forall U\in\mathcal{J}\exists\{B_\alpha\}_{\alpha\in I}\subseteq\mathcal{B}[U=\bigcup_{\alpha\in I}U_\alpha][/ilmath] - all open sets are unions of elements of [ilmath]\mathcal{B} [/ilmath]
Claim: [ilmath]\mathcal{J} [/ilmath] is smallest topology containing [ilmath]\mathcal{B} [/ilmath]
Suppose [ilmath]\mathcal{K} [/ilmath] is another topology on [ilmath]X[/ilmath] and [ilmath]\mathcal{B}\subseteq\mathcal{K} [/ilmath], then:
- [ilmath]\mathcal{J}\subseteq\mathcal{K} [/ilmath]
- Caution:I do not see this - YET
Obviously, you can't just pick some random elements of [ilmath]\mathcal{J} [/ilmath] and call them a basis. That leads to the next theorem:
Theorem: Conditions for a collection of sets to be a basis for a topology
A collection, [ilmath]\mathcal{B} [/ilmath] of sets is a base/basis for some topology [ilmath]\mathcal{J} [/ilmath] on [ilmath]\bigcup\mathcal{B} [/ilmath] if and only if:
- [ilmath]\forall S,T\in\mathcal{B}\forall x\in S\cap T\exists U\in\mathcal{B}[x\in U\subseteq S\cap T][/ilmath]
Terminology: Subbase/subbasis
If [ilmath]\mathcal{A} [/ilmath] is an arbitrary family of sets then [ilmath]\mathcal{B} [/ilmath] - the family of all finite intersections of elements of [ilmath]\mathcal{A} [/ilmath][Note 1] - is a base for a topology, [ilmath]\mathcal{J} [/ilmath] on [ilmath]\bigcup\mathcal{B} [/ilmath].
- We call [ilmath]\mathcal{A} [/ilmath] a subbase/subbasis for [ilmath]\mathcal{J} [/ilmath].
Corollary / Claim: subbase/subbasis conditions
The family [ilmath]\mathcal{A} [/ilmath] is a subbase/subbasis for [ilmath]\mathcal{J} [/ilmath] if and only if:
- [ilmath]\mathcal{A}\subseteq\mathcal{J} [/ilmath] and
- For each member of [ilmath]\mathcal{J} [/ilmath] the member is the union of (finite intersections of elements of [ilmath]\mathcal{A} [/ilmath])
References
Notes
- ↑ Note that the "convention" of taking the intersection of no sets as the entire set (or union of all of the elements of [ilmath]\mathcal{A} [/ilmath]) and the union of no sets as the empty set mean this is okay
