Can a single point on the real line be isolated under any metric space?

Asked Sep 15 '16 at 03:58

Active Sep 15 '16 at 03:58

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My shaky understanding is that whether a single point in a set can be isolated depends on whether a radius can be defined around the point, such that no other point in the set is included in the disc, ball or sphere thus defined.

Is it correct, then, to say that for points on the real line any radius will necessarily include adjacent points, and hence there can't be isolated points?

asked Sep 15 '16 at 03:58

Antoni Parellada

8,889

You can always take the discrete metric, so that every points are isolated. – Sep 15 '16 at 04:00
Can you explain? – Antoni Parellada Sep 15 '16 at 04:01
3

The discrete metric is $$d(x, y) = \begin{cases} 0 & \text{when } x=y \ 1 & \text{if } x\neq y.\end{cases}$$ under this metric all balls with radius $<1$ has only one point. – Sep 15 '16 at 04:03
OK... So we just have to declare a radius of less than one, and by this... rather whimsical measure, we have balls with only one point. – Antoni Parellada Sep 15 '16 at 04:14
And a radius of $1$ would collapse into one single ball the entire real line, like a mathematic magnetar? – Antoni Parellada Sep 15 '16 at 04:17

Can a single point on the real line be isolated under any metric space?

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