We assume that is an infinite set. Cofinite filters The cofinite filter over is the set of subsets of such that is finite. Since is infinite, is not in . It is trivial to check that any superset in of an element of is in , and that the intersection of any two elements of…
Two cases, that I have been unable to unify: a filter base over a power set with a set, or over . First case: filter bases over a power set filter base over , with a set. Definitions Property It is always the case that . Proof: Let be an eventual…
Let and be filters on a same set . Then is also a filter on . Proof: belongs to both and , hence it belongs to , and . Since is not an element of , it is not an element of . Let be an element of and a superset of . Then and…
A category is by definition concrete if there exists a faithful functor from to the category of sets . (A functor is faithful if it is one-to-one (injective) regarding morphisms.) It happens to be the case that in the category , monomorphisms and one-to-one mappings are the same thing; similarly for epimorphisms and surjections. We…
The theorem If and are such that is separately coprime with each , then is coprime with their product . The proof Proof: Through Bézout. being coprime with each , we can write, for each : If we take the product of each of these expressions, we get: Among the terms of this product, all…
For the definition and characterization of the polynomials over a field , see Polynomials over a field. Definition of the degree of a polynomial The degree of a polynomial P, written , is an element of . The symbol is taken to have the usual properties of order and of addition relative to natural numbers….
In this context, we consider a commutative field (simply: field) . The polynomials will be constructed as a certain associative unital algebra over (“-aua”), together with a distinguished element called the formal variable. itself, as a vector space over itself and the usual multiplication in as the third law, is a -aua, Morphisms between two…
Definition Let be a commutative field. The polynomials form a commutative unital algebra over containing an element such that the following universal property holds: For any unital algebra (commutative or otherwise) and any element , there exists exactly one unital algebra morphism such that . If and are elements of , let be the unique…
Definition of a filter Let be a set. Traditional definition of a filter on : A collection of subsets of is a filter on iff: is non-empty (equivalently in context: ); ; ; . It follows from the definition that there can be no filters on the empty set. If we waive the second rule…
In Chapter 31 (“Compact-Open Topology”), the fact that, on the set of the continuous mappings the second topology (the uniform convergence topology) is finer than the first (the compact-open topology) is said to be “clear” but actually needs some non-trivial proof. Part of this is the lemma that I state and prove below. For any …