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Homework Help: Metrizability of a certain topology

  1. Aug 24, 2010 #1

    radou

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    1. The problem statement, all variables and given/known data

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

    3. 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.
     
  2. jcsd
  3. Aug 24, 2010 #2
    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.
     
  4. Aug 24, 2010 #3

    radou

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    Thanks again. :)
     
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