Standard examples
Topological spaces in the following three examples are considered as subspaces of the real line with the standard topology.
- For the set the point 0 is an isolated point.
- For the set each of the points is an isolated point, but 0 is not an isolated point because there are other points in S as close to 0 as desired.
- The set of natural numbers is a discrete set.
In the topological space with topology the element a is an isolated point, even though belongs to the closure of (and is therefore, in some sense, "close" to a). Such a situation is not possible in a Hausdorff space.
The Morse lemma states that non-degenerate critical points of certain functions are isolated.
Two counter-intuitive examples
Consider the set F of points x in the real interval (0,1) such that every digit xi of their binary representation fulfills the following conditions:
- Either or
- only for finitely many indices i.
- If m denotes the largest index such that then
- If and then exactly one of the following two conditions holds: or
Informally, these conditions means that every digit of the binary representation of that equals 1 belongs to a pair ...0110..., except for ...010... at the very end.
Now, F is an explicit set consisting entirely of isolated points but has the counter-intuitive property that its closure is an uncountable set.[1]
Another set F with the same properties can be obtained as follows. Let C be the middle-thirds Cantor set, let be the component intervals of , and let F be a set consisting of one point from each Ik. Since each Ik contains only one point from F, every point of F is an isolated point. However, if p is any point in the Cantor set, then every neighborhood of p contains at least one Ik, and hence at least one point of F. It follows that each point of the Cantor set lies in the closure of F, and therefore F has uncountable closure.