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

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In metrizable spaces, not all open sets can be expressed as countable unions of basis elements. The discussion highlights that while open sets can be represented as unions of open balls defined by a metric, specific topological properties, such as the discrete topology, can prevent this. For instance, in the discrete topology on the real numbers, open sets reduce to singletons or the entire space, leading to uncountable unions for certain sets. Therefore, the assertion does not hold universally across all metrizable spaces.

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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
 

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