what are accumulation points in easy terms? and how do we use them? I've read the definition and have seen videos where they all say the same thing. We have a $\epsilon$ neighborhood around $l$ and if it contains another point, let's say set $A$, other then $l$, then we call $l$ an accumulation point of $A$. But what's the point of this? why do we need accumulation points? and is my description correct?
 A: Here's how I've attempted to conceptualize the notion of an accumulation point and for simplicity let's take the metric case setting.

Definition: Let $(X,d)$ be a metric space. A point $q \in X$ is called an accumulation point of $X$ if for all $\varepsilon > 0$ the set $B(q,\varepsilon) = \{y \in X \mid d(q,y) < \varepsilon\}$ contains a point $p \in X$ such that $q \neq p$.

The idea behind accumulation points is that they are arbitrarily "close" to elements of the $X$ while not necessarily being an element of the set itself. The definition is literally saying that no matter how small a neighbourhood we consider around an accumulation point, we always have at least one point of the set contained in the neighbourhood.
To make the usefulness of accumulation points more clear, it might help to think of the very related definition of an accumulation point of a sequence.

Definition: If $(x_n)$ is a sequence in a metric space, then $x \in X$ is an accumulation point of $(x_n)$ if for every $\varepsilon >0$ the open ball $B(x,\varepsilon)$ contains all but finitely many terms of $(x_n)$.

That is to say an accumulation point of a sequence is a point where infinitely many sequence terms cluster to. The way I think of their usefulness is that if you didn't have an accumulation point, how could you even talk about a sequence $(x_n)$ converging to a point? If we can't get arbitrarily close to that point, i.e if we don't have this accumulation of sequence terms arbitrarily close to the point, how can we start to think about convergence?
Another perspective that you might have already encountered; if you think about general limits of functions, say $f:X \rightarrow \mathbb{R}$ and you want to talk about $\lim_{x \rightarrow c} f(x)$ for this to make sense $c$ must be an accumulation point of the set $X$, because otherwise how can we even think about getting arbitrarily close to the point?
A: Your description is not correct. You say: "We have an epsilon neighborhood". The definition says: "For every epsilon neighborhood".
The answer to "Why do we need them" is not easily summarized here. Suffice it to say, this is one way to think about convergence, closeness, separation, and related ideas. Its value will become apparent as you continue to study topology.
