Hausdorff Convergence (ハウスドルフ収束 / $T_2$) (Definition III.1.13)
- Hausdorff Convergence (ハウスドルフ収束 / $T_2$) (Definition III.1.13) #Card
- A convergence $\xi$ is Hausdorff (or $T_2$) if $\lim_\xi \mathcal{F}$ is at most a singleton for each filter $\mathcal{F}$: ${x_0, x_1} \subset \lim_\xi \mathcal{F} \implies x_0 = x_1$.
A convergence $\xi$ is called Hausdorff (or $T_2$) if $\lim_\xi \mathcal{F}$ is at most a singleton for each filter $\mathcal{F}$, that is, $${x_0, x_1} \subset \lim_\xi \mathcal{F} \implies x_0 = x_1.$$
- Pointwise form: $x_0 \neq x_1 \implies \xi^-(x_0) \cap \xi^-(x_1) = \emptyset$ — 極限の一意性。ただし (Hausdorff) は各点ごとの性質ではない。
- Hausdorff $\implies$ free ($T_1$)。$\iota_X$ は Hausdorff、$\sigma_\mathbb{R}$・$\nu_\mathbb{R}$ も Hausdorff。
- Restrictions and finer convergences of Hausdorff convergences are Hausdorff (Propositions III.1.16, III.3.5); suprema of Hausdorff convergences are Hausdorff, but infima need not be.
位相の場合は T2Space X（極限の一意性は tendsto_nhds_unique）