Isolated point
A set that is made up only of isolated points is called a discrete set (see also discrete space). Any discrete subset "S" of Euclidean space must be countable, since the isolation of each of its points together with the fact that rationals are dense in the reals means that the points of "S" may be mapped into a set of points with rational coordinates, of which there are only countably many. However, not every countable set is discrete, of which the rational numbers under the usual Euclidean metric are the canonical example. A set with no isolated point is said to be dense-in-itself (every neighbourhood of a point contains other points of the set). A closed set with no isolated point is called a perfect set (it has all its limit points and none of them are isolated from it). The number of isolated points is a topological invariant, i.e. if two topological spaces formula_1 and formula_2 are homeomorphic, the number of isolated points in each is equal. Topological spaces in the following three examples are considered as subspaces of the real line with the standard topology. In the topological space formula_6 with topology formula_7, the element formula_8 is an isolated point, even though formula_8 belongs to the closure of formula_10 (and is therefore, in some sense, "close" to formula_11). Such a situation is not possible in a Hausdorff space. The Morse lemma states that non-degenerate critical points of certain functions are isolated. Consider the set formula_12 of points formula_13 in the real interval formula_14 such that every digit formula_15 of their binary representation fulfills the following conditions: 


Informally, these conditions means that every digit of the binary representation of formula_13 which equals 1 belongs to a pair ...0110..., except for ...010... at the very end. Now, formula_12 is an explicit set consisting entirely of isolated points which has the counter-intuitive property that its closure is an uncountable set. Another set formula_12 with the same properties can be obtained as follows. Let formula_30 be the middle-thirds Cantor set, let formula_31 be the component intervals of formula_32, and let formula_12 be a set consisting of one point from each formula_34. Since each formula_34 contains only one point from formula_12, every point of formula_12 is an isolated point. However, if formula_38 is any point in the Cantor set, then every neighborhood of formula_38 contains at least one formula_34, and hence at least one point of formula_12. It follows that each point of the Cantor set lies in the closure of formula_12, and therefore formula_12 has uncountable closure.
Related notions
Another equivalent formulation is: an element "x" of "S" is an isolated point of "S" if and only if it is not a limit point of "S".