some properties of uncountable subsets of the real numbers
Let be an uncountable subset of . Let . For is hereditarily Lindelöff, there is a countable subfamily of such that . For the reason that each of members of has a countable intersection with , we have that is countable. As the open set can be expressed uniquely as the union of its components, and the components are countably many, we label the components as .
Now we’d propose to show that is precisely the set of the points which are unilateral condensation points of .
Let be a unilateral (left, say) condensation point of . So, there is some with countable. So, there is some such that . See, if , then is NOT a condensation point, for has a neighbourhood which has a countable intersection with . But is a condensation point; so, . Similarly, if is a right condensation point, then .
Conversely, each is a left (right, resp) condensation point. Because, for each , we have countable. And as no is in , are condensation points.
So, is the set of non-condensation points - it is countable; and are precisely the unilateral condensation points. So, all the rest are bilateral condensation points. Now we see, all but a countable number of points of are the bilateral condensation points of .
Call the set of all the bilateral condensation points that are IN . Now, take two in . As is a bilateral condensation point of , is uncountable; and as misses atmost countably many points of , is uncountable. So, is a subset of with in-between property.
We summarize the moral of the story: If is an uncountable subset of , then
The points of which are NOT condensation points of , are at most countable.
The set of points in which are unilateral condensation points of , is, again, countable.
The bilateral condensation points of , that are in , are uncountable; even, all but countably many points of are bilateral condensation points of .
The set of all the bilateral condensation points of has got the property: if , then there is also with .
|Title||some properties of uncountable subsets of the real numbers|
|Date of creation||2013-03-22 16:40:42|
|Last modified on||2013-03-22 16:40:42|
|Last modified by||sauravbhaumik (15615)|