$T_0$ Convergence ($T_0$ 収束) (Definition III.1.18)

• $T_0$ Convergence ($T_0$ 収束) (Definition III.1.18) #Card
  • A convergence $\xi$ is $T_0$ if it distinguishes the points: $x_0 \neq x_1 \implies \xi^-(x_0) \neq \xi^-(x_1)$.

Back

A convergence $\xi$ is called $T_0$ if it distinguishes the points, that is, $$x_0 \neq x_1 \implies \xi^-(x_0) \neq \xi^-(x_1).$$

• Each free ($T_1$) convergence is $T_0$（$x_1^\uparrow \in \xi^-(x_1) \setminus \xi^-(x_0)$）.
• The chaotic convergence on a set of cardinality $> 1$ is not $T_0$.
• $T_0$ は制限で保存されない（Example III.7.6、cards/topology/prop-iii-1-16 参照）。Sierpiński 前位相は $T_0$ だが $T_1$ でない例。

Lean

位相の場合は T0Space X