I made this comment in one of my other posts, but because it wasn't the main problem, it was overlooked. I have deleted it from that post (to avoid double posting) and have, evidently, posted it here

I want to clarify the concept of what it means to be "open."

as i said, the definition of an isolated point (see here) bothers me. i was under the impression that a set consisting of a single point was always closed.

an example given in my text (Set Theory and Metric Spaces by Kaplansky) says that the set , in its usual metric as a subset of the real line, is a discrete space with every point (and hence, every subset) is open.

how do you reconcile this concept with the usual definition of being open?