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... presheaf on T is a contravariant functor F : T → C. If U → V is a morphism in T , we call F (i) : F (V ) → F (U ) the restriction map obtained from i. A morphism of presheaves is a natural transformation. To understand this definition, and what it has to do with the more familiar definition of a pre ...
... presheaf on T is a contravariant functor F : T → C. If U → V is a morphism in T , we call F (i) : F (V ) → F (U ) the restriction map obtained from i. A morphism of presheaves is a natural transformation. To understand this definition, and what it has to do with the more familiar definition of a pre ...
A remark on β-locally closed sets
... Proposition 2.1 Every subset A of a topological space (X, τ ) is the intersection of a preopen and a preclosed set, hence pre-locally closed. Proof. Let A ⊆ (X, τ ). Set A1 = A ∪ (X \ cl(A)). Since A1 is dense in X, it is also preopen. Let A2 be the preclosure of A, i.e., A2 = A ∪ cl(int(A)). Clearl ...
... Proposition 2.1 Every subset A of a topological space (X, τ ) is the intersection of a preopen and a preclosed set, hence pre-locally closed. Proof. Let A ⊆ (X, τ ). Set A1 = A ∪ (X \ cl(A)). Since A1 is dense in X, it is also preopen. Let A2 be the preclosure of A, i.e., A2 = A ∪ cl(int(A)). Clearl ...
Algebraic Models for Homotopy Types EPFL July 2013 Exercises 1
... (b) For a map f : X → Y , let M f = IX ∪X Y (pushout of ∂1 and f ). Show that Y → M f is an acyclic cofibration. (c) Assume that Y is also cofibrant and show that the map X → M f (induced by ∂0 ) is a cofibration. 4. (K. Brown’s Lemma) Let F : M → C be a functor from a closed model category to a cat ...
... (b) For a map f : X → Y , let M f = IX ∪X Y (pushout of ∂1 and f ). Show that Y → M f is an acyclic cofibration. (c) Assume that Y is also cofibrant and show that the map X → M f (induced by ∂0 ) is a cofibration. 4. (K. Brown’s Lemma) Let F : M → C be a functor from a closed model category to a cat ...
RECOLLECTIONS FROM POINT SET TOPOLOGY FOR
... We say a collection of open subset N of X containing a point p ∈ X is a neighborhood basis of a point p if for all open sets U that contain p there is a set V ∈ N such that V ⊂ U. We call (X, T ) is first countable if every point has a neighborhood basis consisting of a countable number of sets. Pro ...
... We say a collection of open subset N of X containing a point p ∈ X is a neighborhood basis of a point p if for all open sets U that contain p there is a set V ∈ N such that V ⊂ U. We call (X, T ) is first countable if every point has a neighborhood basis consisting of a countable number of sets. Pro ...
An Introduction to Topology: Connectedness and
... • In a simply connected space any path between two points can be deformed into any other space. • Consider the closed loops, ones in which the starting and ending points are the same. Then they must all be deformable into one another. ...
... • In a simply connected space any path between two points can be deformed into any other space. • Consider the closed loops, ones in which the starting and ending points are the same. Then they must all be deformable into one another. ...
REPRESENTATION THEORY ASSIGNMENT 3 DUE FRIDAY
... is a unipotent group. (Hint: define subgroups of N using subalgebras of n.) (c) Show that every irreducible representation of a unipotent group G is trivial. (Hint: first show that Z acts by a scalar by Schur’s lemma and hence trivially and then continue.) (d) Show that if G is unipotent, then its L ...
... is a unipotent group. (Hint: define subgroups of N using subalgebras of n.) (c) Show that every irreducible representation of a unipotent group G is trivial. (Hint: first show that Z acts by a scalar by Schur’s lemma and hence trivially and then continue.) (d) Show that if G is unipotent, then its L ...
Imagining Negative-Dimensional Space
... Just as the one-dimensional sphere ! ! , the circle, is defined to be the set of points in ℝ! that are a distance of one from the origin, the zero-dimensional sphere ! ! is all the points in ℝ! that are a distance of one from the origin. Therefore ! ! is simply two points, −1, +1 . The above ill ...
... Just as the one-dimensional sphere ! ! , the circle, is defined to be the set of points in ℝ! that are a distance of one from the origin, the zero-dimensional sphere ! ! is all the points in ℝ! that are a distance of one from the origin. Therefore ! ! is simply two points, −1, +1 . The above ill ...
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a ""sufficiently good"" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.