Metrizability of a certain topology

[radou]

Homework Statement

Let U be the topology on R^2 whose subbase is given with the set of all lines in R^2. Is U metrizable?

The Attempt at a Solution

If the set of all lines (let's call it L) in R^2 is a subbase of U, then the family of all finite intersections of L forms a basis for U. Obviously the open sets in this topology are sets of points in R^2. Now, my first thought was that the topology Ud generated by the discrete metric coincides with the topology U. An open ball in the discrete metric space is given with K(x, r) = {y in R^2 satisfying d(x, y) < r}, and its "shape" depends on the value of r. For r <= 1, it is a single point set {x}, and for r > 1, it is R^2.

Unless I'm missing something here, it seems to me that U = Ud, so U is metrizable.

 

[ystael]

You are correct. Every point of the plane is the intersection of some two appropriately chosen lines in the plane, so every single-point set is a basic open set of the topology; hence it must be the discrete topology.

 

[radou]

You are correct. Every point of the plane is the intersection of some two appropriately chosen lines in the plane, so every single-point set is a basic open set of the topology; hence it must be the discrete topology.

Thanks again. :)