Definition of a compact : something I don't get

Question

In my book they define the compact as follows :

$G$ is compact if for every covering of $G$ by open, we can pick out a finite number of the sets that still covers G.

But a union of open is an open.

Thus, if G is compact, there is a finite union of open that covers G. Thus G is open.

But in $\mathbb{R}^n$ compact => closed so I don't get.

PS : I am beginner in topology so I would like a simple explanation using definitions =) (and not a magic theorem that I don't know for example).

Answer 1

Seems like this was cleared up in the comments while I was writing my answer, but here it is anyways (it might still be useful to OP on account of the precise definitions given below).

Definition: An open cover of a set $G$ is a family $\{G_\alpha\}_{\alpha \in \mathcal{A}}$ such that $G \subset \cup_{\alpha \in \mathcal{A}} G_\alpha$.

Definition: A set $G$ is compact if for every open cover $\mathcal{G} = \{G_\alpha\}_{\alpha \in \mathcal{A}}$ of $G$, there exists a finite subset $\{G_{\alpha_n}\}_{n=1}^N$ of $\mathcal{G}$ that is also an open cover of $G$.

Note, in particular, that a set $G$ being compact does not imply that there exists a finite open cover $\{G_{n}\}_{n=1}^N$ of $G$ whose union is exactly equal to $G$ (i.e., $\cup_n G_n = G$). This is the source of confusion.