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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?

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"$A$ is a component" implies that $A$ is connected, but not vice versa. One way to define a connected component is an equivalence class under the relation that $$x\sim y\text{ iff } x \text{ and } y \text{ lie in a connected subspace}.$$ The existence of equivalence classes does not require Zorn's lemma or the Axiom of Choice. A posteriori one can argue that components are indeed maximal connected subspaces, but since you've already proven they exist, you don't need Zorn's Lemma.

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  • $\begingroup$ Very clever. Thanks $\endgroup$ – will hart Nov 20 '13 at 18:08
  • $\begingroup$ But I would still like some clarity on my definition. Here you define connected components differently, but I believe that is just for namesake to show maximal connected subspaces exists w.r.t my definition? $\endgroup$ – will hart Nov 20 '13 at 18:10
  • $\begingroup$ @willhart: your definition is not standard. It would imply that every connected subspace of $\mathbb R^2$ is a component, whereas in the usual definition only $\mathbb R^2$ is a component. $\endgroup$ – Cheerful Parsnip Nov 20 '13 at 18:12

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