The definition of convergence in topology spaces is: Let X be a topology space and a sequence given by $(x_1,x_2,x_3,...) \in X^{\mathbb N}$. A point $x \in X$ is called **limit point** of $(x_n)_{\mathbb N}$ if for every open set U $\in X$ which contains x holds: There exists an index $n_0 \in \mathbb N$ such that $x_n \in U$ $\forall n \geq n_0$.
So in our example we only have one open set which can contain a point $x \in X$, namely X. This is the only (not-empty) open set in our topology space, hence for every limit point in X we got the same open set. And this set contains **all** points of X. Thatswhy every sequence converges.