accumulation points,limit points,cluster points areone and the same thing .Any book in analysis will tell you that

- January 11th 2009, 07:43 AMarchidiDefinitions of accumulation points,limit points,cluster points
accumulation points,limit points,cluster points are

**one and the same thing .Any book in analysis will tell you that** - January 11th 2009, 08:16 AMOpalg
That is demonstrably not true. Unfortunately, authors differ in their definitions of limit point (or accumulation point). Some authors distinguish between a limit of a sequence and a limit point of a set. That must be true in particular for the author of the problem at the start of this thread.

Some (but by no means all) authors define a limit point of a set in such a way as to exclude an isolated point from being a limit point. But when defining a limit point of a sequence, it is essential to allow an isolated point to count as a limit, otherwise a constant sequence would not converge. Lang's definition in comment #3 above refers to limit points of sequences, so of course it allows isolated limit points. I have no idea whether Lang uses the inclusive or the exclusive definition for limit points of sets in metric spaces.

The Wikipedia definition for a limit point of a set is uncompromisingly exclusive: "In mathematics, a**limit point**(or**accumulation point**) of a set*S*in a topological space*X*is a point*x*in*X*that can be "approximated" by points of*S*other than*x*itself." - January 11th 2009, 08:43 AMarchidi
There is agreat difference between the definition

1)of the limit of a sequence and the definition

2) of the**limit point of a sequence.**

**1 IMPLIES 2 and**this is provable as i showed ,while 2 does not imply 1

check the two definition again - January 11th 2009, 12:49 PMmr fantastic