Restriction of a Convergence (収束の制限) (Definition III.1.15)

• Restriction of a Convergence (収束の制限) (Definition III.1.15) #Card
  • If $\xi$ is a convergence on $X$ and $\emptyset \neq V \subset X$, the restriction $\xi|V$ is defined by $\lim{\xi|V} \mathcal{F} := V \cap \lim\xi \mathcal{F}$, where $\mathcal{F}$ is a filter on $V$.

Back

If $\xi$ is a convergence on $X$, and $\emptyset \neq V \subset X$, then the restriction $\xi|V$ of $\xi$ to $V$ is defined by $$\lim{\xi|V} \mathcal{F} := V \cap \lim\xi \mathcal{F},$$ where $\mathcal{F}$ is a filter on $V$（$\mathcal{F}$ は $X$ 上のフィルター基底とみなし、$\lim_\xi \mathcal{F} = \lim_\xi \mathcal{F}^\uparrow$ と読む）.

• Hausdorff・free は制限で保存される（cards/topology/prop-iii-1-16）が、$T_0$ は保存されない（Example III.7.6）。

Lean

位相の場合は部分空間位相 instTopologicalSpaceSubtype（nhds_subtype）