Isolated point

Examples

Standard examples

Topological spaces in the following three examples are considered as subspaces of the real line with the standard topology.

• For the set ${\displaystyle S=\{0\}\cup [1,2],}$ the point 0 is an isolated point.
• For the set ${\displaystyle S=\{0\}\cup \{1,\frac{1}{2},\frac{1}{3},\dots \},}$ each of the points ⁠${\displaystyle \frac{1}{k}}$⁠ 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 ${\displaystyle \mathbb{N}=\{0,1,2,\dots \}}$ of natural numbers is a discrete set.

In the topological space ${\displaystyle X=\{a,b\}}$ with topology ${\displaystyle \tau =\{\mathrm{\varnothing },\{a\},X\},}$ the element a is an isolated point, even though ${\displaystyle b}$ belongs to the closure of ${\displaystyle \{a\}}$ (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 x_i of their binary representation fulfills the following conditions:

• Either ${\displaystyle x_i=0}$ or ${\displaystyle x_i=1.}$
• ${\displaystyle x_i=1}$ only for finitely many indices i.
• If m denotes the largest index such that ${\displaystyle x_m=1,}$ then ${\displaystyle x_{m-1}=0.}$
• If ${\displaystyle x_i=1}$ and ${\displaystyle i<m,}$ then exactly one of the following two conditions holds: ${\displaystyle x_{i-1}=1}$ or ${\displaystyle x_{i+1}=1.}$

Informally, these conditions means that every digit of the binary representation of ${\displaystyle x}$ 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.

Another set F with the same properties can be obtained as follows. Let C be the middle-thirds Cantor set, let ${\displaystyle I_1,I_2,I_3,\dots ,I_k,\dots }$ be the component intervals of ${\displaystyle [0,1]-C}$, and let F be a set consisting of one point from each I_k. Since each I_k 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 I_k, 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.