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 $\sigma$ if and only if:

$\forall U \in \tau_0: \alpha \in U \implies \set {n \in \N: x_n \in U}$ 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 $\sigma$.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $11$. Sierpinski Space: $18$