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Munkres 23.5 is stated as "A space X is called totally disconnected if its only connected subspaces are one-point sets. Show that if X is discrete, then X is totally disconnected. Does the converse hold?"

I'm confused about the definition of totally disconnected. I thought the empty set was trivially a connected subspace of any space X. Wouldn't this violate that X's only connected subspaces are one-point sets? In other words, should this definition also include the empty set?

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  • $\begingroup$ Oftentimes, the reader is supposed to mentally add the word "nonempty" to statements. Here it is meant, of course, "the only non-empty connected subspaces". $\endgroup$ Commented Sep 29, 2021 at 5:18

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It is common to exclude the empty set as an example of a connected space, just as $1$ is excluded from being prime. Munkres probably does this, though I don’t have a copy to hand. If he doesn’t do this, then he means for you to figure out the content of the statement for yourself without being too pedantic.

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