Is every countable metric space separable?

In summary, a topological space is considered separable if it contains a countable dense subset. This means that there is a set with a countable number of elements whose closure is the entire space. In metric spaces, a countable space automatically satisfies this condition as its own subset can be chosen as the dense subset. It is not necessary for the subset to be proper.
  • #1
pivoxa15
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Homework Statement


'In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space.'

http://en.wikipedia.org/wiki/Separable_metric_space

Let (X,d) be a metric space. If X is countable than it immediately satisfies being a separable metric space? Because just choose X itself as the subset. The closure of X must be X. Hence there exists a countable dense subset, namely X itself.

The Attempt at a Solution


Is this correct?

Or they referring to proper subsets only?
 
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  • #2
If they meant proper, they'd say it. Admitting only proper subsets is equivalent to excluding countable sets. There's no good reason for doing that. Also, that article gives an example of a countable space being separable.
 

FAQ: Is every countable metric space separable?

1. What is a separable metric space?

A separable metric space is a mathematical concept that is used to describe a type of topological space. It is a space in which there exists a countable dense subset, meaning that the elements of this subset are "close" to every point in the space. This allows for a more efficient way of studying the space and its properties.

2. How is a separable metric space different from other metric spaces?

A separable metric space differs from other metric spaces in that it has a countable dense subset, whereas other metric spaces may not have this property. This makes separable metric spaces more amenable to analysis and study.

3. What is an example of a separable metric space?

One example of a separable metric space is the set of real numbers with the metric defined as the absolute value of the difference between two numbers. This space is separable because the rational numbers, which are countable, are dense in this space.

4. How is separability related to completeness in a metric space?

In a metric space, completeness refers to the property that every Cauchy sequence converges to a point in the space. A separable metric space can be either complete or incomplete, but if it is complete, then it is also separable. This means that a complete metric space must have a countable dense subset.

5. Why is the concept of separability important in mathematics?

Separable metric spaces are important in mathematics because they allow for a more efficient way of studying and understanding topological spaces. They also have many applications in analysis, topology, and functional analysis, making them a fundamental concept in these fields.

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