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From Topology with Tears, "A topological space $(X,Ƭ)$ is not connected (that is, it is disconnected) if and only if there are non-empty open sets $A$ and $B$ such that $A\cap B = \emptyset$ and $A \cup B = X$"

Thus, I believe the "A topological space $(X,Ƭ)$ is disconnected if it has at least one nonempty, proper subset that is both open and closed" to be an incorrect statement. But what does this statement really propose- could someone provide an example?

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    $\begingroup$ Why do you believe the statement is incorrect? $\endgroup$ – Clayton Dec 21 '16 at 22:03
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    $\begingroup$ I think I made a wrong assumption: whenever a space is made up of a finite number of disjoint connected components like (0,1) and (2,3), the components will be clopen - en.wikipedia.org/wiki/Clopen_set. Thus, this statement is actually correct then... $\endgroup$ – Learner Dec 21 '16 at 22:13
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The statement that you believe incorrect is, in fact, correct. Let $\emptyset \ne Y \subset X$ be closed and open. Since $Y$ is closed and proper, it follows that $X \setminus Y$ is open and nonempty. Therefore $X = Y \cup (X \setminus Y)$, the union of two non-empty, disjoint, open subsets. Then, according to your definition of disconnectedness, $X$ must be disconnected.

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The statement is correct. To see this take proper subset $A\neq\emptyset$ both open and closed. Then since $A$ closed $A^c$ open. Note that it is non-empty because $A$ is proper. Hence, we have two open sets $A,A^C\neq\emptyset$ and $A\cup A^c=X$. This statement propose a quick way to check disconnectedness for a topological space. Once you come up with a proper subset $A$ that is clopen you can conclude disconnectedness.This might be useful in some functional analysis results.

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