@socks I mean you can have a connected clopen set, right? just take the set of all complex numbers with magnitude less than 1, in addition to all complex numbers with magnitude equal to one and positive real part (so, the unit disk, minus its boundary on the left half)

...or is that neither closed nor open? i think it would be either neither or both but i forget how exactly they're defined
-F

@Felthry That is neither closed nor open with Euclidean topology

The DEFINITION of a connected space is one that doesn't have a proper subset that is both closed and open

@socks oh! huh
-F

@Felthry Any subset itself is always going to be both closed and open _in its own topology_, though, by definition of subspace topology. Just not as a subset of the larger space

@socks I don't think I undersatnd the concept of "in its own topology"! but we are too sleepy to really learn about it right now, maybe tomorrow, for now goodnight
-F