TITLE: A topology on the set of lines?
QUESTION [17 upvotes]: Of course any set $X$ can have a topology, but are there more natural topologies, metrics or similar on the set of straight lines in $\mathbb R^2$?

REPLY [4 votes]: Adding to the existing answers - which are quite correct - let me give yet another way of visualizing the space of lines.
Let's first look at the lines in $\mathbb{R}^2$ not passing through the origin; call this set $L_+$. An element $l$ of $L_+$ is specified by an element $\alpha_l$ of the punctured plane $\mathbb{R}^2\setminus\{(0, 0)\}$: namely, $\alpha_l$ is the point on $l$ closest to the origin. To put it another way, draw a line from the origin to $\alpha_l$; the perpendicular to this line, through $\alpha_l$, is $l$. It's easy to check that this is a homeomorphism between $L_+$ (with the subspace topology) and $\mathbb{R}^2\setminus\{(0, 0)\}$.
Now, what happens at the origin? Well, a line through the origin is specified by its angle: specifically, by a point on $S^1$ . . . 
. . . Except that this winds up double-counting lines. E.g. the $x$-axis can be specified by either $(0, 1)$ or $(1, 0)$. So really, the set of lines through the origin looks like $S^1$ with opposite points identified - that is, the projective space $\mathbb{RP}^1$. (This is of course homeomorphic to $S^1$, but that's peculiar to the number $1$.) 
So what description does this give of the space of all lines look like? Well, it looks like the plane $\mathbb{R}^2$, with the origin replaced by a copy of the projective space $\mathbb{RP}^1$. Making this precise is a bit tricky, but a good exercise. And this turns out to be a kind of degenerate example of a crucial construction in algebraic geometry - the blowup (see https://en.wikipedia.org/wiki/Blowing_up).
Incidentally, it's noting that - while this description of the space of lines is correct, and has a number of nice properties - it is misleading in certain ways. In particular, the description sounds non-homogeneous (the origin seems like a special point), while it is clear from thinking about lines in the plane that the space of lines has no distinguished points (and this is clearer from the other descriptions).