Can the union of a closed and an open set (disjoint and separated) be open?

Question

In a topology on X, let A be a closed set disjoint and separated from an open set B (proper subset of X). Is the union of A and B: open? or closed? or open and closed? or not open nor closed? or it is undecidable? or it depends upon the type of topology? or other?

Answer 1

Firstly, I think the definition you are seeking is the following: Two sets of a topological space $X$ are called separated if their closures are disjoint.

Note that a set is open if an only if it is disjoint from its boundary and a set is closed if and only if it contains its boundary. Let $X$ be a topological space, $Z\subseteq X$ closed and $U\subseteq X$ open such that $Z$ and $U$ are separated. It follows from this that $T:=U\cup Z$ satisfies $\partial T = \partial Z\cup\partial U$. Now if $Z$ is not open or $U$ is not closed, then $T$ is neither disjoint from its boundary nor does it contain it, so in these cases $T$ is neither closed nor open.

However, if $U$ is also closed then $T$ will clearly be closed and if $Z$ is also open then $T$ will clearly be open since finite unions preserve both properties.