Irreducible topological space

We say that a topological space $X$ is irreducible if it can not be written as the union of two proper closed subsets.

My question is: Do we consider this definition when the topological space is defined in terms of open subsets or closed subsets? When working with the zariski topology, the definition of irreducible remains the same, in this case the topology is defined in terms of zariski closed subsets.

EDIT: Some clarification on my question. In the picture below, Definition 1.13 tells us when a topological space $X$ is irreducible. Is $X$ defined via open sets or via closed sets in the terminology here https://en.wikipedia.org/wiki/Topological_space? If its defined in terms of open sets, I dont see how the definition translates to the affine n space being irreducible (Example 1.14) since the zariski topology is defined via closed sets.