Topologies making a given ideal nowhere dense or meager
... that 1-yX\X i = K; or (b) there is a compactification yX of X such that I yX\X I = I X I. Notice that there are completely regular noncompact spaces for which (b) in the above fails. For example, it is easy to see that the only compactification of or, considered with the order topology, is its one-p ...
... that 1-yX\X i = K; or (b) there is a compactification yX of X such that I yX\X I = I X I. Notice that there are completely regular noncompact spaces for which (b) in the above fails. For example, it is easy to see that the only compactification of or, considered with the order topology, is its one-p ...
Structure of Hecke algebras of modular forms modulo p
... showed that in this case, all the local components have Krull dimension at least 2 and if ρ̄ is unobstructed, then the corresponding local component is isomorphic to F[[x, y]]. In this paper, we generalize these results with minor changes for an arbitrary prime p > 3 and any level N co-prime to p. ...
... showed that in this case, all the local components have Krull dimension at least 2 and if ρ̄ is unobstructed, then the corresponding local component is isomorphic to F[[x, y]]. In this paper, we generalize these results with minor changes for an arbitrary prime p > 3 and any level N co-prime to p. ...
equivariant homotopy and cohomology theory
... example to keep track of xed point data and to construct new G-spaces from diagrams of potential xed point spaces. Some of the relevant background, such as geometric realization of simplicial spaces and the construction of homotopy colimits, is central to all of algebraic topology. These matters a ...
... example to keep track of xed point data and to construct new G-spaces from diagrams of potential xed point spaces. Some of the relevant background, such as geometric realization of simplicial spaces and the construction of homotopy colimits, is central to all of algebraic topology. These matters a ...
Somewhat continuous functions
... function from (X, 9") onto (X, ^) is somewhat continuous but its restriction to Ä is not somewhat continuous. Theorem 5. If (X, 6^) and (Y, ^) are topological spaces and A is an open subset of X and f : (Л, ^ ^ ) -> (7, ^ ) is a somewhat continuous function such that f {A) is dense in 7, then any e ...
... function from (X, 9") onto (X, ^) is somewhat continuous but its restriction to Ä is not somewhat continuous. Theorem 5. If (X, 6^) and (Y, ^) are topological spaces and A is an open subset of X and f : (Л, ^ ^ ) -> (7, ^ ) is a somewhat continuous function such that f {A) is dense in 7, then any e ...
PhD and MPhil Thesis Classes
... of intrinsic interest. The proof for the topological case gives an effective way to calculate the fundamental groupoid explicitly, whenever we are given a so called discrete covering (Definition 5.1.3). In Section 5.2 we give a big class of such spaces and demonstrate its application. The algebraic ...
... of intrinsic interest. The proof for the topological case gives an effective way to calculate the fundamental groupoid explicitly, whenever we are given a so called discrete covering (Definition 5.1.3). In Section 5.2 we give a big class of such spaces and demonstrate its application. The algebraic ...
Sheaves on Spaces
... to the open U ⊂ X the set F(U ) = {f : U → Y | f is continuous} with the obvious S restriction mappings. We claim that F is a sheaf. To see this suppose that U = i∈I Ui is an open covering, and fi ∈ F(Ui ), i ∈ I with fi |Ui ∩Uj = fj |Ui ∩Uj for all i, j ∈ I. In this case define f : U → Y by setting ...
... to the open U ⊂ X the set F(U ) = {f : U → Y | f is continuous} with the obvious S restriction mappings. We claim that F is a sheaf. To see this suppose that U = i∈I Ui is an open covering, and fi ∈ F(Ui ), i ∈ I with fi |Ui ∩Uj = fj |Ui ∩Uj for all i, j ∈ I. In this case define f : U → Y by setting ...
hohology of cell complexes george e. cooke and ross l. finney
... f i n ite--the closure of each cell is contained in the union of a finite number of (open) cells-- and b) weak topology- the topology on the underlying topological space is the weak topology with respect to the closed cells of the complex.) We g i ve se veral examples of complexes, regular and irre ...
... f i n ite--the closure of each cell is contained in the union of a finite number of (open) cells-- and b) weak topology- the topology on the underlying topological space is the weak topology with respect to the closed cells of the complex.) We g i ve se veral examples of complexes, regular and irre ...
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.