$\mathbb{N}$, with an open set defined as a set with a finite complement, is not path connected.
60700
Continuity of a function to the integers
92571
Sets with no asymptotical density over $\mathbb N$
81927
Example of closed and bounded in $\mathbb Q$ set that is not compact
88595
Prob. 21, Sec. 17 in Munkres' TOPOLOGY, 2nd ed: Closure and complementation can result in at most 14 different sets
148507
Why an open set in $R^n$ might not be written as countable union of disjoint open intervals?
4518
How would one go about proving that the rationals are not the countable intersection of open sets?
79424
Don't understand proof that interior of a set is open
67279
Is the union of finitely many open sets in an omega-cover contained within some member of the cover?
147534
The set of irrational numbers is not a $F_{\sigma}$ set.
77957
How is $\mathbb N$ actually defined?
132382
Showing a set is nowhere dense in $C([0,1])$
114990
Is the complement of countably many disjoint closed disks path connected?
113915
A star shaped open set in $\mathbb{R}^n$ is diffeomorphic to $\mathbb{R}^n$
75970
$\mathbb Z[1/2]$ is not finitely generated?
85225
The set of integers is not open or is open
5230
Why can a 1 element set be a member of another set but not a subset of it?
85586
Proving $\mathbb{Q}$ isn't homeomorphic to $\mathbb{Z}$