Convergence in Indiscrete Space

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Theorem

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

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


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


Proof

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

Hence:

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

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

$\blacksquare$


Sources