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 $T$?