Isolated Point (孤立点) (Definition III.1.12)
- Isolated Point (孤立点) (Definition III.1.12) #Card
- An element $x$ of $X$ is $\xi$-isolated if $x \in \lim_\xi \mathcal{F}$ implies $\mathcal{F} = x^\uparrow$, that is, $\xi^-(x) = {x^\uparrow}$.
An element $x$ of $X$ is called $\xi$-isolated (or simply isolated) if $x \in \lim_\xi \mathcal{F}$ implies that $\mathcal{F} = x^\uparrow$; equivalently $\xi^-(x) = {x^\uparrow}$ — no filter “from outside of $x$” converges to $x$.
- A convergence is discrete $\iff$ all elements of the underlying set are isolated.
- 非孤立点が高々 1 つの収束が prime（cards/topology/prime-convergence）。