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

From Maths
Jump to: navigation, search
(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)
Line 10: Line 10:
 
*# {{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>
−
Note that we could also say:
+
<noinclude>
−
* 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]}} 
+
−
** 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>
+
 
==Notes==
 
==Notes==
 
<references group="Note"/>
 
<references group="Note"/>

Revision as of 21:14, 15 January 2017

Grade: A
This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
I could do this now but I can't be bothered!

Statement

Let [ilmath]X[/ilmath] be a set and let [ilmath]\mathcal{B}\in\mathcal{P}(\mathcal{P}(X))[/ilmath] be any collection of subsets of [ilmath]X[/ilmath], then:

  • [ilmath](X,\{\bigcup\mathcal{A}\ \vert\ \mathcal{A}\in\mathcal{P}(\mathcal{B})\})[/ilmath] is a topological space with [ilmath]\mathcal{B} [/ilmath] being a basis for the topology [ilmath]\{\bigcup\mathcal{A}\ \vert\ \mathcal{A}\in\mathcal{P}(\mathcal{B})\}[/ilmath]

if and only if

  • we have both of the following conditions:
    1. [ilmath]\bigcup\mathcal{B}=X[/ilmath] (or equivalently: [ilmath]\forall x\in X\exists B\in\mathcal{B}[x\in B][/ilmath]) and
    2. [ilmath]\forall U,V\in\mathcal{B}\ \forall x\in U\cap V\ \exists W\in\mathcal{B}[x\in W\subseteq U\cap V][/ilmath][Note 1]

Notes

  1. ↑ We could of course write:
    • [ilmath]\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)][/ilmath]

References


Retrieved from "https://wiki.unifiedmathematics.com/index.php?title=Topology_generated_by_a_basis/Statement&oldid=3941"