Application of $'$, the derived set operation If $F_1=\{1/2, 3/4, 7/8, 15/16,\dots, 1\}$, then $1$ is a limit point. In Stillwell's "Real Numbers," p. 128, he says the $'$ operation (derived set, or set of limit points) can be performed exactly twice to $F_1$ since each of its points except $1$ are isolated.
Also he has a picture with the caption: "The set $F_1$ with the derived set $\{1\}$".
My question, please, is what are the two distinct performances of $'$, as I would think one application would eliminate all the points other than $1$.
Thanks
 A: Apply it once:  $F_1' = ${limit points of $F_1$} = $ \{1\}$.
Apply it twice:  $F_1'' = ${ limit points of $F'$} = {limit points of $\{1\}$} = $\emptyset$.
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Worth noting that isolated points (points which have neighborhoods containing no other points of the set) and limit points (points of which every neighborhood contain other points of the set) are exact opposite concepts.  No isolated point is  a limit point and no limit point is an isolated point.
So all isolated points will be removed by the $'$ operation.
But once removed the resulting derived set is a different set and what was previously a limit point may not be in the new set.  In this case all the points in the neighborhood of $1$ were isolated points of $F_1$ and were removed.  As a result in the derived set all the former neighborhoods have no points of the new set.
One thing that used to throw me off all the time was I had the mistaken idea that interior points and limit points were somehow opposites.  They are not.  In fact if the space is such that open neighborhoods must contain more than one point (i.e. isolated points are not open sets) then interior points are always limit points.  (But limit points need not be interior points.)  My error was that I was confusing limit points with boundary points.  Boundary points are limit points of the set and also limit points of the compliment of the set and thus a specific type, and not general, limit points.  (On the other hand, interior points [which in the euclidean topology are limit points] and boundary points [which are always a type of limit point] are opposite types of limit points.)
okay... that was a big diversion.
