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…

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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…

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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….

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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…

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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…

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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 …

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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…

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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…

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Connected subsets one of which meets the closure of the other

Exercise 224 (Chapter 32, “Connectedness”): Let and be connected subsets of topological space such that intersects . Prove that is connected. Find an example to show that it is not enough to assume that intersects . Let us assume that and are open subsets of such that is empty and that . To show that…

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Union of transitively overlapping connected spaces

Exercise 225 (Chapter 32, “Connectedness”): Prove the following generalization of theorem 40. Let ( in ) be a collection of connected subsets of topological space X, and let the equivalence relation on this collection of sets generated by if have just one equivalence class. Then is connected. For clarity, rather than a family I will…

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