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

[jem05]

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.

 

[Office_Shredder]

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