Notes:Basis for a topology/McCarty

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:

• [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] - 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:

1. [ilmath]\mathcal{A}\subseteq\mathcal{J} [/ilmath] and
2. For each member of [ilmath]\mathcal{J} [/ilmath] the member is the union of (finite intersections of elements of [ilmath]\mathcal{A} [/ilmath])

Applications to continuity

Theorem

Suppose [ilmath]f:X\rightarrow Y[/ilmath] is a function between two topological spaces, [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath]. Let [ilmath]\mathcal{A} [/ilmath] be a sub-basis for [ilmath]\mathcal{K} [/ilmath], then:

• [ilmath]f[/ilmath] is continuous if and only if [ilmath]\forall A\in\mathcal{A}[f^{-1}(A)\in\mathcal{J}][/ilmath]

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TODO: I really need to create the pages that show the pre-image of functions preserves things like unions and intersections

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