Free Convergence (自由収束 / $T_1$) (Definition III.1.13)
- Free Convergence (自由収束 / $T_1$) (Definition III.1.13) #Card
- A convergence $\xi$ is free (or $T_1$) if $\lim_\xi x^\uparrow \subset {x}$ for each $x$.
A convergence $\xi$ is said to be free (traditionally called $T_1$) if $$\lim_\xi x^\uparrow \subset {x}.$$
- By (centered) the inclusion is in fact an equality.
- Each Hausdorff convergence is free; each free convergence is $T_0$.
- 特徴づけ: $\xi$ is free $\iff$ $x \in \lim_\xi \mathcal{F} \implies \operatorname{ker} \mathcal{F} \subset {x}$（cards/topology/prop-iii-1-14）。
- Being free is a pointwise property: $\xi$ is free $\iff$ $\xi^-(x) \setminus {x^\uparrow}$ consists of free filters for each $x$.