Intro to Categories
... U ∈ Obj[C] is a universally repelling object in C if for any A ∈ Obj[C] there exists a unique morphism f : U → A. It is also called an initial object A universally repelling object, therefore, is an object that maps uniquely to every other object in the category. The word unique here is very importa ...
... U ∈ Obj[C] is a universally repelling object in C if for any A ∈ Obj[C] there exists a unique morphism f : U → A. It is also called an initial object A universally repelling object, therefore, is an object that maps uniquely to every other object in the category. The word unique here is very importa ...
QUOTIENT SPACES – MATH 446 Marc Culler
... topology on P is the collection T = {O ⊂ P | ∪O is open in X}. Thus the open sets in the quotient topology are collections of subsets whose union is open in X. We can think of the partition elements as “fat points”, and the open sets as collections of “fat points” whose union is open as a subset of ...
... topology on P is the collection T = {O ⊂ P | ∪O is open in X}. Thus the open sets in the quotient topology are collections of subsets whose union is open in X. We can think of the partition elements as “fat points”, and the open sets as collections of “fat points” whose union is open as a subset of ...
Topological Vector Spaces and Continuous Linear Functionals
... that for each x 6= 0 there exists a ν such that ρν (x) > 0. For each y ∈ X and each index ν, define gy,ν (x) = ρν (x − y). Then X, equipped with the weakest topology making all of the gy,ν ’s continuous, is a topological vector space, i.e., is Hausdorff and addition and scalar multiplication are con ...
... that for each x 6= 0 there exists a ν such that ρν (x) > 0. For each y ∈ X and each index ν, define gy,ν (x) = ρν (x − y). Then X, equipped with the weakest topology making all of the gy,ν ’s continuous, is a topological vector space, i.e., is Hausdorff and addition and scalar multiplication are con ...
