Intersection of filters
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…
Concrete categories: injective/surjective implies mono/epi; functor generalization
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…
Coprimality with a product of integers
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…
Degree and Euclidean division of polynomials
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….
Polynomials over a field
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…
The composition of polynomials
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…
Definitions concerning filters
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…
Margin between a compact subspace and an including open subspace
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 …
Open tubes
In Chapter 31, “The Compact-Open Topology”, it is asserted that “clearly” the third topology on (with and the real line) is finer than the second. The issue is more fully discussed here. In this post I discuss the concept of open tubes, particular subsets of the Cartesian product with a set and a metric space. Definition…
Uniform convergence topology and the open set topology
Chapter 31, “The Compact-Open Topology”, describes the compact-open topology, and goes on to compare it with two other topologies on in the case where . It asserts that “clearly” the second of these three is finer than the first, and the third finer than the second. This may well be “clear” intuitively, but is not…