I wanted to give some additional information to complement Vhailor's answer.
It is true that a set is closed if and only if it contains all of its limit points. However, you can't define what it means to be a limit point (or what it means for a sequence to converge) unless you have a notion of an open set.
A topological space is a set and the additional information of which subsets are considered open sets, subject to certain axioms. In general, any set has many different topologies. There are a few general ones worth mentioning.
Let $X$ be a set, possibly with additional structure (which we shall specify below). The following are all natural topologies on $X$
The trivial topology The only open sets are $X$ and $\emptyset$.
The discrete topology Every subset of $X$ is open, and hence closed.
The finite complement topology The closed sets in $X$ (other than $X$ itself) are the finite subsets of $X$.
The order topology If $X$ is a linearly ordered set, we can declare all open intervals of the form $(a,b)=\{x\in X \mid a< x < b\}$ to be open. Then, all unions of these are also open. Edit: we also want to include intervals of the form $(a,\infty)=\{x\in X \mid a< x \}$ and $(\infty,b)= \{x\in X \mid x < b\}$. Otherwise, the endpoints would give us the wrong topology, e.g., on $[0,1]$.
The metric topology If $X$ has a metric, we declare all open balls $B_r(p)=\{x\in X \mid d(x,p) to be open. Just as with the order topology, we must take all unions of these balls to get a topology.
The first three examples are somewhat strange if you haven't seen them before, and the first two tend to be more for the sake of counterexamples than anything else.
The last two will give you all the examples of topological spaces you saw before you saw the definition of a topology.
For example, $\mathbb{R}$ is both a linearly ordered set and a metric space. Therefore, there are at least two natural topologies we could think to put on $\mathbb{R}$. In fact, they coincide.
More generally, $\mathbb{R}^n$ can be made into a metric space in many different ways. If we define the $\ell_p$ norm by $||(x_1, \ldots x_n)||_p=\left(\sum |x_i|^p \right)^{1/p}$, this norm generates a metric. Each of these metrics generates a topology. However, all these topologies are the same. In fact, all metrics on $\mathbb{R}^n$ coming from linear norms generates the same topology. This does not happen for infinite dimensional vector spaces.
In summary, there are lots of good choices for a topology on a space, and you must fix one before you can talk about things like closed sets, convergence, or most other things considered to be topological.