Convergence in Indiscrete Space
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Theorem
Let $\struct {S, \set {S, \O} }$ be an indiscrete space.
Let $\sequence {x_n}$ be any sequence in $S$.
Then $\sequence {x_n}$ 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
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $4$: The Hausdorff condition: $4.1$: Motivation: Example $4.1.3$