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?
Best Answer
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.