Difference Between Limit Point and Accumulation Point? I want to clarify the definition of limit point and accumulation point.
According to many of my text books they are synonymous that is $x$ is a limit/accumulation point of set $A$ if open ball $B(x, r)$ contains an an element of $A$ distinct from $x$.
But from one of the problems in Aksoy: A Problem Book in Real Analysis says:
Show that if $x \in (M,d)$ is an accumulation point of $A$, then $x$ is a limit point
of $A$. Is the converse true?
So what is the definition?

From the Wikipedia article on "accumulation points":

Accumulation point is a type of limit point.
types of limit points are:
if every open set containing x contains infinitely many points of S then x is a specific type of limit point called an ω-accumulation point of S.
If every open set containing x contains uncountably many points of S then x is a specific type of limit point called a condensation point of S.
If every open set U containing x satisfies $|U ∩ S| = |S|$ then x is a specific type of limit point called a complete accumulation point of S.
A point $x \in X$ is a cluster point or accumulation point of a sequence $(x_n)_n \in N$ if, for every neighbourhood V of x, there are infinitely many natural numbers $n$ such that $x_n \in V$.This is equivalent to the assertion that $x$ is a limit of some subsequence of the sequence $(x_n)_n \in N$.

