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

Question

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?

Answer 1

Yes, by the definition of topology an arbitrary union of open sets is open.

In particular, if each singleton is open all sets are open and the topology is discrete.

Answer 2

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.

Answer 3

If every singleton is an open set the topology is discrete. Indeed, every subset of the topological space is open (this follows from the definition of topological space) and this is the definition of discrete topology.