A metric space having a countable dense subset has a countable base.

In summary, the student is trying to find a solution to a homework problem and is having trouble understanding why one needs to choose rational radius. The rational numbers are chosen precisely because they're countable.
  • #1
rumjum
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1. Homework Statement

Every separable metric space has countable base, where base is collection of sets {Vi} such that for any x that belongs to an open set G (as subset of X), there is a Vi such that x belongs to Vi.

2. Homework Equations

Hint from the book of Rudin: Center the point in a countable dense subset of the metric space and have a union of all "rational" radius.

3. The Attempt at a Solution

I need to understand this problem a bit more, so would appreciate any hints. I am mainly unclear as to why one needs to choose rational radius. Is it because rational numbere are countable? So we can have finitely many Vi.

My understanding so far is that if x belongs to G , then we need to prove that G is a union of Vi , where i belongs to N.

Now, based on the hint, we can find a point "p" in the dense subset A Intersection X, and have neighborhoods of all rational radii. Let any such neighborhood be Vi. Any neighborhood is an open set and the union of such open sets is open. Let this set be G , then any element that belongs to G , belongs to Union of Vi. But Vi is a collection of rational radii neighborhood and rational numbers are countable. Hence, the collection Vi is countable. In other words, {Vi} is the base.

Can anyone comment?
 
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  • #2
The rational numbers are chosen precisely because they're countable. You can choose balls of radius 1/n for n in N if that makes you happy (and this is actually cleaner).

Then you said that we have finitely many V_i. Of course this is completely wrong. I'll let you figure out why.

Your understanding about G being a union of V_i's is correct. Although i doesn't have to belong to N: any union is OK (although by the nature of the collection of V_i's, any such union is going to be countable).

Your last paragraph isn't convincing. What you're supposed to do is start with a countable collection of open sets {V_i}. Then show that for any given open set G in X, we can choose a suitable subcollection of {V_i}, say {V*_i} such that G is the union of the V*_i's.
 
  • #3
I suppose I wanted to say that the collection of Vi is countable and not finitely many. Anyways...
 

What is a metric space?

A metric space is a mathematical structure consisting of a set of objects, or points, and a function that measures the distance between any two objects in the set. This function is called a metric and must satisfy certain properties, such as non-negativity, symmetry, and the triangle inequality.

What is a countable dense subset?

A countable dense subset of a metric space is a subset of points that is both countable (meaning its elements can be put into a one-to-one correspondence with the natural numbers) and dense (meaning that every point in the space is either in the subset or a limit point of the subset).

What is a countable base?

A countable base of a metric space is a collection of open sets that can be used to generate all other open sets in the space. This means that any open set in the space can be expressed as a union of sets from the countable base.

Why does a metric space having a countable dense subset imply a countable base?

This is because a countable dense subset provides enough points to generate a countable base. By using the points in the dense subset, we can construct open sets that cover the entire space and therefore generate a countable base.

What are the real-world implications of a metric space having a countable dense subset and a countable base?

This fact has many applications in mathematics and other fields, such as computer science and physics. In mathematics, it allows for the study of topological properties of metric spaces using countable structures, which can simplify proofs and calculations. In computer science, it can be used in algorithms for data compression and optimization. In physics, it can be applied to the study of fractals and other complex systems.

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