View source for Notes:Basis for a topology/New terminology

==Definitions==
Here we will use '''GBasis''' for a generated topology (by a basis) and '''TBasis''' for a basis of an existing topology.
===GBasis===
Let {{M|X}} be a [[set]] and {{M|\mathcal{B}\subseteq\mathcal{P}(X)}} be a collection of [[subset of|subsets of]] {{M|X}}. Then we say:
* {{M|\mathcal{B} }} is a '''GBasis''' if it satisfies the following 2 conditions:
*# {{M|\forall x\in X\exists B\in\mathcal{B}[x\in B]}} - every element of {{M|X}} is contained in some GBasis set.
*# {{M|\forall B_1,B_2\in\mathcal{B}\forall x\ B_1\cap B_2\exists B_3\in\mathcal{B}[B_1\cap B_2\ne\emptyset\implies(x\in B_3\subseteq B_1\cap B_2)]}}<ref group="Note">Note that {{M|x\in B_3\subseteq B_1\cap B_2}} is short for:
* {{M|x\in B_3\wedge B_3\subseteq B_1\cap B_2}}</ref><ref group="Note">Note that if {{M|B_1\cap B_2}} is empty (they do not [[intersection|intersect]]) then the [[logical implication]] is true regardless of the RHS of the {{M|\implies}}} sign, so we do not care if we have {{M|x\in B_3\wedge B_3\subseteq B_1\cap B_2}}! Pick any {{M|x\in X}} and aany {{M|B_3\in\mathcal{B} }}!</ref>
Then {{M|\mathcal{B} }} ''induces'' a [[topology]] on {{M|X}}.

Let {{M|\mathcal{J}_\text{Induced} }} denote this topology, then:
* {{M|1=\forall U\in\mathcal{P}(X)\big[U\in\mathcal{J}_\text{Induced}\iff\big(\forall p\in U\exists B\in\mathcal{B}[p\in B\wedge B\subseteq U]\big)\big]}}
** {{Caution|{{M|\forall p\in U\exists B\in\mathcal{B}[p\in B\wedge B\subseteq U]}} is actually just {{M|1=\bigcup_{p\in U}(\text{the }B\text{ that exists})=U}} really. This is more often seen as "If a '''GBasis''' then the collection of all unions is a topology"}}
===TBasis===
Suppose {{Top.|X|J}} is a [[topological space]] and {{M|\mathcal{B}\subseteq\mathcal{P}(X)}} is some collection of subsets of {{M|X}}. We say:
* {{M|\mathcal{B} }} is a '''TBasis''' if it satisfies both of the following:
*# {{M|\forall B\in\mathcal{B}[B\in\mathcal{J}]}} - all the basis elements are themselves open.
*# {{M|1=\forall U\in\mathcal{J}\exists\{B_\alpha\}_{\alpha\in I}[\bigcup_{\alpha\in I}B_\alpha=U]}}
If we have a '''TBasis''' for a topological space then we may talk about its open sets differently:
* {{M|1=\forall U\in\mathcal{P}(X)\big[U\in\mathcal{J}\iff\big(\forall p\in U\exists B\in\mathcal{B}[p\in B\wedge B\subseteq U]\big)\big]}} ''This is called the '''Basis Criterion'''''
==Notes==
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