Indiscrete Non-Singleton Space is not T0
Jump to navigation
Jump to search
Theorem
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space which has more than one element.
Then $T$ is not a $T_0$ (Kolmogorov) space.
Proof
Let $a, b \in S$.
By definition of indiscrete space, $S$ is the only open set in $T$ which is non-empty.
So (trivially) there is no open set in $T$ containing $a$ and not $b$, or $b$ and not $a$.
Hence the result, by definition of $T_0$ (Kolmogorov) space.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $4$. Indiscrete Topology: $10$