Currently, an adherent point is a closure point (Adamson, 1995). An adherent point which is not a limit point is isolated. But this does not explain what the term "isolated" means.
L.T. Adamson, A General Topology Workbook, Birkhauser, 1995.
Let us first agree on some définitions: An adherent point of a set A, say x, is any closure point of A, which means that every neighbourhood of x meets A at least at one point which may be x if the latter belongs to A (Notice that x need not belong to A). This is different, of course, from a cluster point, which is : x is a cluster point of A if every neighbourhood of x meets A at least at one point different from x (again x need not belong to A).
Now, an isolated adherent point, I think, is a closure point which is not a closter point. Therefore x must belong to A but there is at least one neighbourhood of x which meets A exactly at x. This gives also that the singleton whose only element is x is an open set in the relative topology of A.
Martin and Lahbib,
Great posts! Many thanks for your observations.
Here is another view of isolated points: A point x of a topological space X is called an isolated point of a subset S of X if x belongs to S and there exists in X a neighborhood of x not containing other points of S. This is equivalent to saying that the singleton {x} is an open set in the topological space S. For example,
0 is an isolated point in $S = {0} \cup [1,2]$.
http://en.wikipedia.org/wiki/Isolated_point
You may find the attached paper interesting:
M. Gonzalez, M. Mbekhta, M. Oudghiri, On the isolated points on the surjective spectrum of a bounded operator, Proc. Amer. Math. Soc. 136 (10), 2008, 3521-3528;
see, for example, Section 3, p. 3525, on Riesz points.