Uniformities and uniformly continuous functions on locally
... A topological group G is said to be a SIN-group if the left and the right uniform structures on G coincide. Every precompact topological group and every Abelian topological group is a SIN-group; the linear Lie group SL2(R) provides the best known specimen of a topological group that is not SIN. (See ...
... A topological group G is said to be a SIN-group if the left and the right uniform structures on G coincide. Every precompact topological group and every Abelian topological group is a SIN-group; the linear Lie group SL2(R) provides the best known specimen of a topological group that is not SIN. (See ...
Problem Set #1 - University of Chicago Math
... B. Give an example to show that S may not be a topology on X. C. Prove that S forms a sub-basis for a topology on X and that this topology is the smallest topology on X that contains all of the Tα . VII. Let X be a set. A. Prove that if B is a basis for a topology on X, then the topology generated b ...
... B. Give an example to show that S may not be a topology on X. C. Prove that S forms a sub-basis for a topology on X and that this topology is the smallest topology on X that contains all of the Tα . VII. Let X be a set. A. Prove that if B is a basis for a topology on X, then the topology generated b ...
Homomorphisms - Columbia Math
... Proposition 2.1. Let G1 and G2 be groups and let f : G1 → G2 be a homomorphism. Then (i) If H1 ≤ G1 , the f (H1 ) ≤ G2 . In other words, the image of a subgroup is a subgroup. (ii) If H2 ≤ G2 , the f −1 (H2 ) ≤ G1 . In other words, the preimage of a subgroup is a subgroup. Proof. (i) We need to chec ...
... Proposition 2.1. Let G1 and G2 be groups and let f : G1 → G2 be a homomorphism. Then (i) If H1 ≤ G1 , the f (H1 ) ≤ G2 . In other words, the image of a subgroup is a subgroup. (ii) If H2 ≤ G2 , the f −1 (H2 ) ≤ G1 . In other words, the preimage of a subgroup is a subgroup. Proof. (i) We need to chec ...
Finite group schemes
... Let G/k be a group scheme over some field k. Let G0 denote the connected component of G that contains e. One expects that G0 is a subgroup scheme of G. This is indeed true. One needs to prove that the image of G0 ×k G0 ⊂ G ×k G under the multiplication map m : G ×k G → G is contained in G0 . We are ...
... Let G/k be a group scheme over some field k. Let G0 denote the connected component of G that contains e. One expects that G0 is a subgroup scheme of G. This is indeed true. One needs to prove that the image of G0 ×k G0 ⊂ G ×k G under the multiplication map m : G ×k G → G is contained in G0 . We are ...
Isbell duality. - Mathematics and Statistics
... 1.1. Notation. We will be considering a number of special classes of monic and epic morphisms. All our categories are concrete (with obvious underlying functors) and we Ä will use  / only for maps that, up to equivalences, are embeddings, algebraic as well as topological (when a topology is present ...
... 1.1. Notation. We will be considering a number of special classes of monic and epic morphisms. All our categories are concrete (with obvious underlying functors) and we Ä will use  / only for maps that, up to equivalences, are embeddings, algebraic as well as topological (when a topology is present ...
Homology stratifications and intersection homology Geometry & Topology Monographs Colin Rourke Brian Sanderson
... It is not hard to see, from graphical considerations, that V –shaped permutations are precisely the same as allowable permutations. Thus the singular permutation homology groups for allowable permutations are precisely the intersection homology groups. Further it can be seen that, given an X̄ –allow ...
... It is not hard to see, from graphical considerations, that V –shaped permutations are precisely the same as allowable permutations. Thus the singular permutation homology groups for allowable permutations are precisely the intersection homology groups. Further it can be seen that, given an X̄ –allow ...
PDF version - University of Warwick
... It is not hard to see, from graphical considerations, that V –shaped permutations are precisely the same as allowable permutations. Thus the singular permutation homology groups for allowable permutations are precisely the intersection homology groups. Further it can be seen that, given an X̄ –allow ...
... It is not hard to see, from graphical considerations, that V –shaped permutations are precisely the same as allowable permutations. Thus the singular permutation homology groups for allowable permutations are precisely the intersection homology groups. Further it can be seen that, given an X̄ –allow ...
Pseudocompactness and uniform continuity in topological groups
... This work contains a number of theorems about pseudocompact groups. Our first and most useful theorem allows us to decide whether or not a given (totally bounded) group is pseudocompact on the basis of how the group sits in its Weil completion. A corollary, which permits us to answer a question pose ...
... This work contains a number of theorems about pseudocompact groups. Our first and most useful theorem allows us to decide whether or not a given (totally bounded) group is pseudocompact on the basis of how the group sits in its Weil completion. A corollary, which permits us to answer a question pose ...
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied both through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.