全近傍系は基本近傍系

証明

$ \forall x\in X\forall N:

$ N\in\mathcal N(x)

$ \iff N\in\mathcal N(x)\land N\subseteq N

$ \implies\exist N^*\in\mathcal N(x):N^*\subseteq N

$ \implies\forall x\in X\forall N\in\mathcal N(x)\exist N^*\in\mathcal N(x):N^*\subseteq N

$ \underline{\forall x\in X:\mathcal N(x)\in\mathscr N^*(x)\quad}_\blacksquare