# When $S$, the set of straight lines in $\mathbb{R}^2$, is a subbasis for a topology, what is the topology?

Salam. What are your thoughts on this? Again, please provide as much detail in your explanation as possible...

Let $S$ be the collection of all straight lines in the plane $\Bbb R^2$. If $S$ is a subbasis for a topology $T$ on the set $\Bbb R^2$, what is the topology?

These are my thoughts so far:

If $S$ is a subbasis then the basis it generates is all of the finite intersections of all of the straight lines in $\Bbb R^2$. Let's call this basis $B$. The topology $T$ is generated by basis $B$.

In order to get an idea of what $T$ is let's look at what kind of sets are in $B$. We know that any straight line is in the basis since any straight line is its intersection with itself. But what do finite intersections of lines look like? Are they just all the points in the plane? If that is so then what is $T$?

Thanks.

• You should include detail on what you are having problems with and attempts made so that the people here can help you the best they can. – Paul Plummer Mar 19 '15 at 7:18
• Whenever I see something is put on hold by zarathustra...I cannot help but think "Thus spoke Zarathustra." – Daniel W. Farlow Mar 19 '15 at 10:39

I think that Scott is right. But building up from your answer, I think that the open sets will look like any collection of points with any assortment of lines

• ... or, more simply, the discrete topology. – Simon Rose Mar 19 '15 at 13:32
• @SimonRose Yes, the discrete topology! Thanks! – user224530 Mar 19 '15 at 14:02
• Also, the title is wrong. It's all straight lines, I don't know why I included something about the origin. I'll try to fix it. – user224530 Mar 19 '15 at 14:03