# If in a given topological space X every set $\{x\}, x \in X,$ is open, is then $X$ discrete

If in a given topological space $$X$$ every set $$\left\{x\right\}, x \in X,$$ is open, is then $$X$$ discrete? My question is this: is an infinite union of open sets open? Or is it only valid for a countable union? I get that the definition of a discrete space is that every subset is open, but does it go the other way?

• Yes. Every union of open sets is open. It's in the definition of the topology. And yes, discrete $\Longleftrightarrow$ every set is open. – Dog_69 Feb 5 at 20:54

The answer is positive: $$X$$ is discrete, once very set $$Y=\{a,b,c,...\}$$ can be formed as an union of open sets $$\{a\},\{b\},...$$- and, therefore, is open. And yes, infinite- countable or not!- unions of open sets are open. Only intersections are required to be finite in the definition of a topology.