Structures closed under unions
Slides of a talk I gave on this topic
Mathematical structures that are closed under countable or uncountable unions often give rise to a maximal structure generated by some set. We would go through examples in topology to explore this idea. This is closely related to structures closed under intersection and similar ideas in topology.
Interiors
From topology, arbitrary unions of open sets are open by definition.
As such for any subset \(A\) of a topological space \(X\) we can define the interior of \(A\) to be
\[\text{Int}(A)=\bigcup_{\substack{F \text{ open in } X \\ F \subset A}} F\]and it satisfies the following properties

\(\text{Int}(A) \subset A\)

\(\text{Int}(A)\) open

If \(F\) open and \(F \subset A\), then \(F \subset \text{Int}(A)\)
I.e. \(\text{Int}(A)\) is the largest open set that contains \(A\)
Connected Components
Again in topology, you can define the connected component for some element \(a\) in some topological space \(X\) to be
\[C_a = \bigcup_{\substack{S \text{ connected} \\ a \in S}} S\]and it satisfies the following properties

\(a \in C_a\)

\(C_a\) connected (Union of connected sets with nonempty intersection is connected)

If \(S\) connected and \(a \in S\), then \(S \subset C_a\)
I.e. \(C_a\) is the largest connected set that contains \(a\)
Going Further
You could generalise the idea of the interior and define an interior operator / algebra. This is closely related to the closure operator discussed in another blog post on structures closed under intersections.