Determine all the connected subsets of $\mathbb{Z}.$?
I thinks $\mathbb{N} $ will be the all connected subset of $\mathbb{Z}.$
I'm right or wrong
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3Under what topology? – Randall Oct 27 '19 at 19:49
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1@Randall usual topology – jasmine Oct 27 '19 at 19:49
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2Possible duplicate of What are all of the connected subsets of $\mathbb{Q},\Bbb Z$? – Dietrich Burde Oct 27 '19 at 19:50
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1In this case that typically means discrete, in which case this is trivial. – Randall Oct 27 '19 at 19:53
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That duplicate post doesn't say anything about $\mathbb Z$, even in its title.... – Oct 27 '19 at 20:06
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@MatthewDaly well, instead of saying singletons in $\Bbb Q$, the answer now says singletons in $\Bbb Z$. Note $\Bbb N$ is not connected. – Mirko Oct 27 '19 at 20:21
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Every singleton $\{x\}$ in every discrete space (in particular $\Bbb Z$ with its usual topology) is both open and closed. Hence the component of connectedness of $x$ is $\{x\}$.
Mirko
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