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 $(X,\mathcal{ J })$ be a topological space, let $\mathcal{B}\subseteq\mathcal{J}$. We say $\mathcal{B}$ is a base or basis for $\mathcal{J}$ if:

• $\forall U\in\mathcal{J}\exists\{B_\alpha\}_{\alpha\in I}\subseteq\mathcal{B}[U=\bigcup_{\alpha\in I}U_\alpha]$ - all open sets are unions of elements of $\mathcal{B}$

Claim: $\mathcal{J}$ is smallest topology containing $\mathcal{B}$

Suppose $\mathcal{K}$ is another topology on $X$ and $\mathcal{B}\subseteq\mathcal{K}$, then:

• $\mathcal{J}\subseteq\mathcal{K}$
  • Caution:I do not see this - YET

Obviously, you can't just pick some random elements of $\mathcal{J}$ 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, $\mathcal{B}$ of sets is a base/basis for some topology $\mathcal{J}$ on $\bigcup\mathcal{B}$ if and only if:

• $\forall S,T\in\mathcal{B}\forall x\in S\cap T\exists U\in\mathcal{B}[x\in U\subseteq S\cap T]$

Terminology: Subbase/subbasis

If $\mathcal{A}$ is an arbitrary family of sets then $\mathcal{B}$ - the family of all finite intersections of elements of $\mathcal{A}$ - is a base for a topology, $\mathcal{J}$ on $\bigcup\mathcal{B}$.

• We call $\mathcal{A}$ a subbase/subbasis for $\mathcal{J}$.

Corollary / Claim: subbase/subbasis conditions

The family $\mathcal{A}$ is a subbase/subbasis for $\mathcal{J}$ if and only if:

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

Applications to continuity

Theorem

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

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

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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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