Domain Theory - School of Computer Science, University of
... function on a complete lattice, or more generally on a directed-complete partial order with least element, has a least fixpoint. (For an account of the history of this result, see [LNS82].) Some early uses of this result in the context of formal language theory were [Ard60, GR62]. It had also found ...
... function on a complete lattice, or more generally on a directed-complete partial order with least element, has a least fixpoint. (For an account of the history of this result, see [LNS82].) Some early uses of this result in the context of formal language theory were [Ard60, GR62]. It had also found ...
On Noether`s Normalization Lemma for projective schemes
... scheme D+ (f ) = Spec B(f ) for any principal open subset D+ (f ) ⊆ X . It can fp = M(p) , for every p ∈ Proj A, where M(p) is the be proved as before that M set of elements degree 0 of Mp . This last two examples show us more clearly the connection between modules over a ring and sheaves on a ringe ...
... scheme D+ (f ) = Spec B(f ) for any principal open subset D+ (f ) ⊆ X . It can fp = M(p) , for every p ∈ Proj A, where M(p) is the be proved as before that M set of elements degree 0 of Mp . This last two examples show us more clearly the connection between modules over a ring and sheaves on a ringe ...
Definitions of compactness and the axiom of choice
... It follows from (∗) that ù \ (S0 ∪ ... ∪ Sk ∪ {nk+1 }) is infinite. Since m is finite, it follows that for some i, 0 ≤ i ≤ m, ù \ (S0 ∪ ... ∪ Sk ∪ Ti ) is infinite, say i = p. Thus, ù \ (S0 ∪ ... ∪ Sk ∪ Tp ) is infinite and we can take Sk+1 = Tp . It follows by induction that the set {S0 , S1 , · · ...
... It follows from (∗) that ù \ (S0 ∪ ... ∪ Sk ∪ {nk+1 }) is infinite. Since m is finite, it follows that for some i, 0 ≤ i ≤ m, ù \ (S0 ∪ ... ∪ Sk ∪ Ti ) is infinite, say i = p. Thus, ù \ (S0 ∪ ... ∪ Sk ∪ Tp ) is infinite and we can take Sk+1 = Tp . It follows by induction that the set {S0 , S1 , · · ...
Metrization Theorem
... In this theorem we will show that regularity of X and existence of a countably locally finite basis for X are equivalent to metrizability. The proof of these condition imply merizability is very closed to the second proof of the Uryshon metrization Theorem. ...
... In this theorem we will show that regularity of X and existence of a countably locally finite basis for X are equivalent to metrizability. The proof of these condition imply merizability is very closed to the second proof of the Uryshon metrization Theorem. ...
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.