« first day (1668 days earlier)   

11:46 AM
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Q: Is my definition correct? How do I prove the finite union of elements of $\mathscr U$ lie in $\mathscr U$?

Math geekDefinition of basis for a topology on $X$ given in the Foundation of Topology By Patty given by Let $(X,\mathscr T)$ be a topological space. A basis for $\mathscr T$ is a subcollection $\mathscr B$ of $\mathscr T$ with the property that if $U\in \mathscr T$ then $U=\emptyset$ or there is ...

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Q: Show that $\{Z(p):p\in \mathscr P\}$ is a basis for the closed sets of some topology (Called the Zariski topology) on $\mathbb R^n$.

Math geekLet $n\in \mathbb N$, and let $\mathscr P$ denote the collection of all polynomials in $n$ variables. For $p\in \mathscr P$, let $Z(p)=\{(x_1,x_2,...,x_n)|p(x_1,x_2,...,x_n)=0\}$. Show that $\{Z(p):p\in \mathscr P\}$ is a basis for the closed sets of some topology (Called the Zariski topology) on...

Please help me.
 
 
3 hours later…
2:44 PM
It seems to me that the notion of basis for closed sets seems to be used only seldom.
@Mathgeek Did you want to write there "subcollection $\mathscr D$ of $\mathscr U$" rather than "subcollection $\mathscr D$ of $\mathscr T$"?
 
 
1 hour later…
4:00 PM
@MartinSleziak I don't understand. where should I change?
 
4:11 PM
"Let $(X,\mathscr U)$ be a family of closed sets(closed sets of topology on $X$). A basis for $\mathscr U$ is a subcollection $\mathscr D$ of $\mathscr T$ with the property that ..."
@Mathgeek This is what your post says right after "my attempt".
If you say that $\mathscr D$ is a subcollection of $\mathscr T$, then you're actually saying that it consists of open sets.
I suppose that you want to say that $\mathscr D$ consist of closed sets.
 
 
1 hour later…
5:22 PM
@MartinSleziak Thank you very much I have corrected it :)
 

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