I have just come across the idea of a connected component of a topological space. And firstly I would just like some clarity on the definition, as it seemed a little vague. Here is what I understand the precise definition to be:
Given a topological space $X$ then $A \subset X$ is a connected component of $X$ iff $A$ is connected in its subspace topology.
Hopefully this definition is correct. My main question is that can one conclude that $X$ is the disjoint union of maximal connected components (ordered by inclusion) iff Zorn's lemma is assumed? I.e to get that maximal connected components exists, do we need Zorn's lemma?