1. ## Concept Clarification

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 $\displaystyle X = \{ 1, \frac 12, \frac 13, \cdots , \frac 1n, \cdots \}$, 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?

2. Originally Posted by Jhevon
(Set Theory and Metric Spaces by Kaplansky) says that the set $\displaystyle X = \{ 1, \frac 12, \frac 13, \cdots , \frac 1n, \cdots \}$, 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?
Are you confusing $\displaystyle X = \{ 1, \frac 12, \frac 13, \cdots , \frac 1n, \cdots \}$ the subset of $\displaystyle R^1$ with $\displaystyle X$ as a metric space by itself? As a subset of $\displaystyle R^1$, no subset of $\displaystyle X$ is open. But as a space in and of itself every subset of $\displaystyle X$ is open.

To reconcile this idea think of the ball centered at 1/4 with radius 1/25. Is that a subset of {1/4}? Is that not the definition of an open set? So {1/4} is an open set in the metric space $\displaystyle X$.

3. Originally Posted by Plato
To reconcile this idea think of the ball centered at 1/4 with radius 1/25. Is that a subset of {1/4}? Is that not the definition of an open set? So {1/4} is an open set in the metric space $\displaystyle X$.
that's the thing. technically i would think it is not a subset of {1/4}. the ball would be outside the point. the only part of the ball that has anything in common with {1/4} is the center.

i take it that we consider the ball to be a subset of {1/4} if its radius does not surpass half the distance between {1/4} and its closest neighbors? if so, why wouldn't the book mention a concept like that. it would seem that they expect it to be obvious to the reader, but it's not, at least not to me. i guess it was the concept of {1/4} as a subset of R^1 that was confusing me. metric spaces is something new to me after all, not too sure how to navigate through that world yet

4. Originally Posted by Jhevon
the ball would be outside the point. the only part of the ball that has anything in common with {1/4} is the center.
But that is exactly the point. In that discrete space 1/4 is the only point in the ball centered at 1/4 with radius 1/25. Every point is isolated. Each singleton set is both open and closed.