Metrizability of a certain topology

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In summary, the conversation discusses the topology U on R^2 with a subbase of all lines in R^2. The question is whether this topology is metrizable. The discussion concludes that U is metrizable and is equivalent to the discrete topology.
  • #1
radou
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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.
 
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  • #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.
 
  • #3
ystael said:
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. :)
 

1. What is metrizability in topology?

Metrizability is a property of a topological space that means it can be equipped with a metric that induces the same topology as the original one. This means that the open sets in the metric topology will be the same as the open sets in the original topology.

2. How is metrizability related to continuity?

In topology, continuity is a fundamental concept that measures how well-behaved a function is with respect to the underlying topology. A function is continuous if the preimage of any open set is open. Metrizability is important because it allows us to define continuity in terms of a metric, making it easier to work with and understand.

3. What are some examples of metrizable topologies?

The most well-known example of a metrizable topology is the Euclidean topology on n-dimensional Euclidean space. Other common examples include the discrete topology, the lower limit topology, and the product topology on countably infinite Cartesian products of real numbers.

4. How is metrizability tested for a given topology?

To test for metrizability, we can use a variety of criteria such as the Urysohn metrization theorem, the Nagata-Smirnov metrization theorem, or the Bing metrization theorem. These theorems provide necessary and sufficient conditions for a topology to be metrizable.

5. What are the implications of a topology being non-metrizable?

If a topology is non-metrizable, it means that there is no metric that can induce the same topology. This can make it more difficult to work with and understand the topological space. Additionally, some important theorems and concepts in topology, such as the Baire category theorem, only apply to metrizable spaces. However, there are still many interesting and useful non-metrizable topologies that are studied in mathematics.

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