Every metric space is Hausdorff

In summary, the theorem typically assumes that the topology of the metric space is generated by the metric itself. However, if a different topology is used, such as the trivial topology, the space may not be Hausdorff even though the metric remains unchanged. Is this due to a mistake or simply a lack of precision in the statement of the theorem?"
  • #1
Dead Boss
150
1
The usual proof of this theorem seems to assume that the topology of the metric space is the one generated by the metric. But if I use another topology, for example the trivial, the space need not be Hausdorff but the metric stays the same. Am I missing something or is the statement of the theorem just sloppy?
 
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  • #2
If you say that you have a metric space [itex](X,d)[/itex], then it is always assumed that we take the topology generated by the metric. If we happen to take another topology, then we always state this explicitely.

It's the same as saying that you work with [itex]\mathbb{R}^n[/itex]. The topology is always assumed to be Euclidean unless otherwise stated.
 
  • #3
Thanks. :smile:
 

1. What is a metric space?

A metric space is a set of points with a defined distance function, called a metric, that measures the distance between any two points in the set.

2. What does it mean for a metric space to be Hausdorff?

A metric space is Hausdorff if for any two distinct points in the space, there exists open sets that contain one point but not the other. In other words, every point in a Hausdorff space has a neighborhood that does not contain any other points.

3. Why is it important for a metric space to be Hausdorff?

Hausdorff spaces have important properties that make them useful in many areas of mathematics. For example, they allow for unique limits of sequences and continuous functions, and they are essential for the development of topological spaces.

4. How can you prove that every metric space is Hausdorff?

The proof involves showing that the Hausdorff property holds for the standard metric on any given metric space. This can be done by considering the open balls around each point and showing that they do not intersect.

5. Can a non-Hausdorff metric space exist?

Yes, a non-Hausdorff metric space can exist, but it would not have many of the useful properties that Hausdorff spaces have. In fact, many common spaces, such as the real line with the lower limit topology, are not Hausdorff.

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