To be specific, let " M " be a smooth manifold and let " O ( M ) " be the category of open subspaces of " M " i . e . the category where the objects are the open subspaces of " M ", and the morphisms are inclusion maps.
32.
The resolvent of a quintic is of degree 6 this corresponds to an exotic inclusion map as a transitive subgroup ( the obvious inclusion map fixes a point and thus is not transitive ) and, while this map does not make the general quintic solvable, it yields the exotic outer automorphism of S 6 see automorphisms of the symmetric and alternating groups for details.
33.
The resolvent of a quintic is of degree 6 this corresponds to an exotic inclusion map as a transitive subgroup ( the obvious inclusion map fixes a point and thus is not transitive ) and, while this map does not make the general quintic solvable, it yields the exotic outer automorphism of S 6 see automorphisms of the symmetric and alternating groups for details.
34.
For instance, the inclusion map \ iota : Z \ rightarrow R from the integers to the real line ( with the integers equipped with the cofinite topology ) is continuous when restricted to an integer, but the inverse image of a bounded open set in the reals with this map is at most a finite number of points, so not open in " Z ".
35.
Equivalent to this definition, a space " X " is semi-locally simply connected if every point in " X " has a neighborhood " U " for which the homomorphism from the fundamental group of U to the fundamental group of " X ", induced by the inclusion map of " U " into " X ", is trivial.
36.
The short and ( somewhat ) informal answer to your first question is this : An arbitrary subset of a topological space can be equipped with a topology such that the corresponding inclusion map is continuous ( i . e . a morphism in the concrete category of topological spaces ), whereas an arbitrary subset of a group cannot necessarily be equipped with a group structure such that the corresponding inclusion map is a homomorphism ( i . e . a morphism in the concrete category of groups ).
37.
The short and ( somewhat ) informal answer to your first question is this : An arbitrary subset of a topological space can be equipped with a topology such that the corresponding inclusion map is continuous ( i . e . a morphism in the concrete category of topological spaces ), whereas an arbitrary subset of a group cannot necessarily be equipped with a group structure such that the corresponding inclusion map is a homomorphism ( i . e . a morphism in the concrete category of groups ).
38.
In this case, some authors call " R " the " direct sum of the rings " R " " i " " and write, but this is incorrect from the point of view of category theory, since it is usually not a coproduct in the category of rings : for example, when two or more of the " R " " i " are nonzero, the inclusion map fails to map 1 to 1 and hence is not a ring homomorphism.
