Convergence in Indiscrete Space

Theorem

Let $\left({A, \left\{{A, \varnothing}\right\}}\right)$ be an indiscrete space.

Let $\left \langle {x_n} \right \rangle$ be any sequence in $A$.

Then $\left \langle {x_n} \right \rangle$ converges to any point $x$ of $A$.

Proof

For any open set $U \subseteq A$ such that $x \in T$, we must have $U = A$.

Hence $\forall n \ge 1: x_n \in U$.

The result follows from definition of a convergent sequence in a topological space.

$\blacksquare$

Note

This demonstrates that in the general topological space it is not necessarily the case that a Sequence has One Limit at Most.