Difference between revisions of "Topology generated by a basis/Statement"

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(Created page with "<noinclude> {{Requires references|grade=A|msg=I could do this now but I can't be bothered!}} __TOC__ ==Statement== </noinclude>Let {{M|X}} be a set and let {{M|\mathcal{B}...")
 
(Moving note section)
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*# {{M|1=\forall U,V\in\mathcal{B}\ \forall x\in U\cap V\ \exists W\in\mathcal{B}[x\in W\subseteq U\cap V]}}<ref group="Note">We could of course write:
 
*# {{M|1=\forall U,V\in\mathcal{B}\ \forall x\in U\cap V\ \exists W\in\mathcal{B}[x\in W\subseteq U\cap V]}}<ref group="Note">We could of course write:
 
* {{M|1=\forall U,V\in\mathcal{B}\ \forall x\in \bigcup\mathcal{B}\ \exists W\in\mathcal{B}[(x\in U\cap V)\implies(x\in W\wedge W\subseteq U\cap V)]}}</ref>
 
* {{M|1=\forall U,V\in\mathcal{B}\ \forall x\in \bigcup\mathcal{B}\ \exists W\in\mathcal{B}[(x\in U\cap V)\implies(x\in W\wedge W\subseteq U\cap V)]}}</ref>
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Note that we could also say:
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<noinclude>
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* Let {{M|\mathcal{B} }} be a collection of [[sets]], then {{M|(\bigcup\mathcal{B},\{\bigcup\mathcal{A}\ \vert\ \mathcal{A}\in\mathcal{P}(\mathcal{B})\})}} is a [[topological space]] {{iff}} {{M|1=\forall U,V\in\mathcal{B}\ \forall x\in U\cap V\ \exists W\in\mathcal{B}[x\in W\subseteq U\cap V]}} 
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** This is just condition {{M|2}} from above, clearly {{M|1}} isn't needed as {{M|1=\bigcup\mathcal{B}=\bigcup\mathcal{B} }} (obviously/trivially)<noinclude>
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==Notes==
 
==Notes==
 
<references group="Note"/>
 
<references group="Note"/>

Revision as of 21:14, 15 January 2017

Grade: A
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Statement

Let X be a set and let B∈P(P(X)) be any collection of subsets of X, then:

  • (X,{⋃A | A∈P(B)}) is a topological space with B being a basis for the topology {⋃A | A∈P(B)}

if and only if

  • we have both of the following conditions:
    1. ⋃B=X (or equivalently: ∀x∈X∃B∈B[x∈B]) and
    2. ∀U,V∈B ∀x∈U∩V ∃W∈B[x∈W⊆U∩V][Note 1]

Notes

  1. Jump up ↑ We could of course write:
    • ∀U,V∈B ∀x∈⋃B ∃W∈B[(x∈U∩V)⟹(x∈W∧W⊆U∩V)]

References


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