Pavement (敷石 / ペイヴメント) (Definition III.2.2)

• Pavement (敷石 / ペイヴメント) (Definition III.2.2) #Card
  • A collection of filters $\mathbb{D} \subset \xi^-(x)$ is a pavement of $\xi$ at $x$ if for each filter $\mathcal{F} \in \xi^-(x)$ there is $\mathcal{D} \in \mathbb{D}$ such that $\mathcal{F} \supset \mathcal{D}$.

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A collection of filters $\mathbb{D} \subset \xi^-(x)$ is called a pavement of $\xi$ at $x$ if for each filter $\mathcal{F} \in \xi^-(x)$, there is $\mathcal{D} \in \mathbb{D}$ such that $\mathcal{F} \supset \mathcal{D}$.

• $x$ に収束するフィルター全体を「下から支える」収束フィルターの族。
• $\xi$ is a pretopology $\iff$ each point has a pavement consisting of one filter（cards/topology/prop-iii-2-3）。
• $\sigma_\mathbb{R}$ の pavement は各点で必ず非可算（cards/topology/sigma-pavement-uncountable）。