Zero is Accumulation Point of Sequence in Sierpiński Space

Theorem
Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.

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

has $0$ as an accumulation point.

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

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

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