Zero is Accumulation Point of Sequence in Sierpiński Space

Theorem
Let $T = \left({\left\{{0, 1}\right\}, \tau_0}\right)$ be a Sierpiński space.

The sequence in $T$:
 * $\sigma = \left\langle{0, 1, 0, 1, \ldots}\right\rangle$

has $0$ as an accumulation point.

Proof
By definition, $\alpha$ is an accumulation point of $\sigma$ :
 * $\forall U \in \tau_0: \alpha \in U \implies \left\{{n \in \N: x_n \in U}\right\}$ is infinite.

Both $\left\{{0}\right\}$ and $\left\{{0, 1}\right\}$ contain $0$, which occurs an infinite number of times in $\sigma$.

Hence, by definition, $0$ is an accumulation point of $\sigma$.