Editing Isolated point (section)

===Two counter-intuitive examples===
Consider the set {{mvar|F}} of points {{mvar|x}} in the real interval {{math|(0,1)}} such that every digit {{mvar|x{{sub|i}}}} of their [[binary number|binary]] representation fulfills the following conditions: 
* Either <math>x_i=0</math> or <math>x_i=1.</math>
* <math>x_i=1</math> only for finitely many indices {{mvar|i}}.
* If {{mvar|m}} denotes the largest index such that <math>x_m=1,</math> then <math>x_{m-1}=0.</math>
* If <math>x_i=1</math> and <math>i < m,</math> then exactly one of the following two conditions holds: <math>x_{i-1}=1</math> or <math>x_{i+1}=1.</math>
Informally, these conditions means that every digit of the binary representation of <math>x</math> that equals 1 belongs to a pair ...0110..., except for ...010... at the very end.

Now, {{mvar|F}} is an explicit set consisting entirely of isolated points but has the counter-intuitive property that its [[Closure (topology)|closure]] is an [[uncountable set]].<ref>{{Citation|last=Gomez-Ramirez|first=Danny|title=An explicit set of isolated points in R with uncountable closure|journal = Matemáticas: Enseñanza universitaria |publisher=Escuela Regional de Matemáticas. Universidad del Valle, Colombia|volume = 15|year = 2007|pages = 145–147|url=http://www.redalyc.org/articulo.oa?id=46815211}}</ref>

Another set {{mvar|F}} with the same properties can be obtained as follows.  Let {{mvar|C}} be the middle-thirds [[Cantor set]], let <math>I_1,I_2,I_3,\ldots,I_k,\ldots</math> be the [[Connected_space#Connected_components|component]] intervals of <math>[0,1]-C</math>, and let {{mvar|F}} be a set consisting of one point from each {{mvar|I{{sub|k}}}}.  Since each {{mvar|I{{sub|k}}}} contains only one point from {{mvar|F}}, every point of {{mvar|F}} is an isolated point. However, if {{mvar|p}} is any point in the Cantor set, then every neighborhood of {{mvar|p}} contains at least one {{mvar|I{{sub|k}}}}, and hence at least one point of {{mvar|F}}.  It follows that each point of the Cantor set lies in the closure of {{mvar|F}}, and therefore {{mvar|F}} has uncountable closure.