Metrizable Space: Open Sets as Countable Unions of Basis Elts?

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In summary, the question asks if any open set in a metrizable space can be written as a countable union of basis elements. The answer is no, as illustrated by the example of the real numbers with the discrete topology. However, certain properties such as being T1 may make this assertion true.
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jem05
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hello,
i have a clarification question:
if i have a metrizable space, can any open set be written as a countable union of basis elts?
if not, can it have some properties that make this assertion true, like being T1 or sth like that.
thanks a lot.

just to clarify why i think so, i have a space X with a topology T on it.
Let's say this topology is metrizable, with a metric d. Then, the open sets O(x,r) = {y in X , d(x,y)<r} form a basis for this topology. so any elt in this topology, t in T can be written as a countable union of such O's.
I just want to know if this is works.
thx.
 
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The answer is no. Take the real numbers with the discrete topology: the distance between any two points is 1. Then consider the set of all positive numbers. Any open ball is one of two things: a single point or all the real numbers. So we're stuck with writing the positive numbers as an uncountable union of singletons
 

1. What is a metrizable space?

A metrizable space is a topological space that satisfies the first countability axiom, meaning that every point in the space has a countable neighborhood basis. This allows for the definition of a metric, or a distance function, on the space, which can be used to define open sets.

2. What is the significance of open sets being countable unions of basis elements?

This property is significant because it allows for a more precise and concrete definition of open sets in a metrizable space. It also ensures that the open sets will be well-behaved and follow the axioms of a topological space.

3. How are basis elements related to open sets in a metrizable space?

Basis elements are fundamental building blocks of open sets in a metrizable space. They are used to generate open sets through countable unions, and every open set can be written as a union of basis elements. This provides a more manageable way to understand and work with open sets in the space.

4. Can a topological space be metrizable if it is not first countable?

No, a topological space must satisfy the first countability axiom to be metrizable. This axiom ensures that there is a countable basis for the topology, which is necessary for defining a metric on the space.

5. How does the concept of metrizable space relate to other topological concepts?

Metrizable spaces are a special type of topological space that have a metric defined on them. They are closely related to other concepts such as Hausdorff spaces, compact spaces, and separable spaces. In fact, all metric spaces are also metrizable spaces, but not all metrizable spaces are metric spaces.

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