seminar notes - Andrew.cmu.edu
... The product topology on the product space is the topology generated by the sets {p−1 α (U ) | α ∈ A and U ⊆ Xα is open} The only fact about the product topology that we will need for this talk is that a sequence (x(n) )∞ n=1 converges in the product topology if and only if the sequence (n) (xα )∞ co ...
... The product topology on the product space is the topology generated by the sets {p−1 α (U ) | α ∈ A and U ⊆ Xα is open} The only fact about the product topology that we will need for this talk is that a sequence (x(n) )∞ n=1 converges in the product topology if and only if the sequence (n) (xα )∞ co ...
The Bryant--Ferry--Mio--Weinberger construction of generalized
... N .X / RL of an embedding X RL , for some large L. One can assume that N .X / is a mapping cylinder neighborhood (see Lacher [5, Corollary 11.2]). The global Poincaré duality of Poincaré spaces does not imply the local homology condition (ii) above. The local homology condition can be understo ...
... N .X / RL of an embedding X RL , for some large L. One can assume that N .X / is a mapping cylinder neighborhood (see Lacher [5, Corollary 11.2]). The global Poincaré duality of Poincaré spaces does not imply the local homology condition (ii) above. The local homology condition can be understo ...
Cardinal properties of Hattori spaces on the real lines and their superextensions
... topology is called the Vietoris topology. The expX with the Vietoris topology is called the exponential space or the hyperspace of X [8]. Let X be a T1 -space. Denote by expn X the set of all closed subsets of X cardinality of that is not greater than the cardinal number n, i.e. expn X = {F ∈ expX : ...
... topology is called the Vietoris topology. The expX with the Vietoris topology is called the exponential space or the hyperspace of X [8]. Let X be a T1 -space. Denote by expn X the set of all closed subsets of X cardinality of that is not greater than the cardinal number n, i.e. expn X = {F ∈ expX : ...
Notes
... “space of candidates” for the positions that the system may attain, and given an initial condition, the task is to find out which position the system actually will attain at a given time. We shall usually assume that the set Q is a smooth manifold, so that tools from analysis can be used to study th ...
... “space of candidates” for the positions that the system may attain, and given an initial condition, the task is to find out which position the system actually will attain at a given time. We shall usually assume that the set Q is a smooth manifold, so that tools from analysis can be used to study th ...
A few results about topological types
... Definition 27 (generated σ-algebra) Let U be a set and S ⊆ P(U) ; σ(S) is the generated σ-algebra on U from S if and only if it is the intersection of all the σ-algebras on U that contain S. Definition 28 (Borel σ-algebra) Let (U, TU ) be a topological space ; the Borel σ-algebra on (U, TU ), noted ...
... Definition 27 (generated σ-algebra) Let U be a set and S ⊆ P(U) ; σ(S) is the generated σ-algebra on U from S if and only if it is the intersection of all the σ-algebras on U that contain S. Definition 28 (Borel σ-algebra) Let (U, TU ) be a topological space ; the Borel σ-algebra on (U, TU ), noted ...
Algebraic approach to p-local structure of a finite group: Definition 1
... (I) If P ≤ S is fully normalized in F, then P is also fully centralized and AutS (P ) is a Sylow subgroup of (AutF (P )). (II) If P ≤ S and ϕ ∈ HomF (P, S) are such that ϕP is fully centralized, then ϕ extends to ϕ̄ ∈ HomF (Nϕ, S), where Nϕ = {g ∈ NS (P ) | ϕ ◦ cg ◦ ϕ−1 ∈ AutS (ϕP )}. Condition (I) ...
... (I) If P ≤ S is fully normalized in F, then P is also fully centralized and AutS (P ) is a Sylow subgroup of (AutF (P )). (II) If P ≤ S and ϕ ∈ HomF (P, S) are such that ϕP is fully centralized, then ϕ extends to ϕ̄ ∈ HomF (Nϕ, S), where Nϕ = {g ∈ NS (P ) | ϕ ◦ cg ◦ ϕ−1 ∈ AutS (ϕP )}. Condition (I) ...
On Maps and Generalized Λb-Sets
... since for the g.Λb -set {b} of (Y, σ), the inverse image f −1 ({b}) = {b} is not a g.Λb-set of (X, τ ). Theorem 3.6. A map f : (X, τ ) → (Y, σ) is g.Λb-irresolute (resp. g.Λb continuous) if and only if, for every g.Λb-set A (resp. closed set A) of (Y, σ) the inverse image f −1 (A) is a g.Vb -set of ...
... since for the g.Λb -set {b} of (Y, σ), the inverse image f −1 ({b}) = {b} is not a g.Λb-set of (X, τ ). Theorem 3.6. A map f : (X, τ ) → (Y, σ) is g.Λb-irresolute (resp. g.Λb continuous) if and only if, for every g.Λb-set A (resp. closed set A) of (Y, σ) the inverse image f −1 (A) is a g.Vb -set of ...
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.