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Notes on Nonstandard Analysis
02-27-2021, 12:38 AM (This post was last modified: 03-05-2021 12:47 PM by elim.)
Post: #1
Notes on Nonstandard Analysis
-- for the Working Mathematician -- 2nd Ed
Peter A. Loeb, And Manfred P.H. Wolff

1.1.1 Def. $\;\mathcal{U}\subset\mathscr{P}(\mathbb{N})$ is a free ultrafilter in $\mathbb{N}$ if
$(1)\quad\varnothing\not\in\mathcal{U}$
$(2)\quad (A\in\mathcal{U})\wedge(B\in\mathcal{U})\implies A\cap B\in\mathcal{U}$
$(3)\quad(A\subset\mathbb{N})\cap(A\not\in\mathcal{U})\implies \mathbb{N}-A\in\mathcal{U}$
$(1\sim 3)$ make$\,\mathcal{U}$ an ultrafilter$:\;\mathcal{U}\ni A\subset B\implies B\in\mathcal{U}$
Proof.$\quad(\mathcal{U}\ni A\subset B\not\in\mathcal{U})\implies\mathcal{U}\ni A\cap B^{\,c}=\varnothing$
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