By Alexandre Grothendieck, Jean Alexandre Dieudonne

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When f : X → Y is an inclusion map, and y is an ultrafilter on Y with X ∈ y , then y | X is an ultrafilter on X . (b) For every x ∈ X , the principal filter x˙ = ↑{x} is an ultrafilter on X . (c) If X is an ultrafilter on the set β X = {x | x is an ultrafilter on X } , then X is an ultrafilter on X . 2 Proposition Every proper filter a on X is contained in an ultrafilter x on X . P. In fact, this statement can be used to formulate a formally finer assertion as follows. 3 Corollary For a filter b and a proper filter a on X such that a there is an ultrafilter x on X with a ⊆ x but b ⊆ x .

In fact, key results (such as the equivalence of the open-set and the ultrafilter-convergence presentations of topologies) rely on it. We alert the reader to each new use of the Axiom of Choice by putting the symbol © in the margin. The symbol merely indicates our use of Choice at the instance in question, without any affirmation that the use is actually essential. II Monoidal structures Gavin J. Seal and Walter Tholen This chapter provides a compactly written introduction to the order- and categorytheoretic tools most commonly used throughout the remainder of the book.

The filters are those A which satisfy (1) x, y ∈ A =⇒ x ∧ y ∈ A, (2) ∈ A, and (3) x ∈ A, x ≤ y =⇒ y ∈ A for all x, y ∈ Z . If Z has a bottom element ⊥, properness then means (4) ⊥ ∈ / A. Every down-directed set A in Z generates the filter ↑ A in Z , in which case A is also called a filter base for ↑ A. In particular, for every element a ∈ Z , one has the principal filter ↑ a in Z . The dual notions are those of up-directed set, ideal, proper ideal, ideal base, and principal ideal. We use these notions predominantly for Z = P X (for some set X ), ordered by inclusion.