Connes–Karoubi long exact sequence for Fréchet sheaves
... Examples. (a) Let X ′ be an integral noetherian scheme of finite type over C and let X be a closed integral subscheme of X ′ , corresponding to a sheaf I of ideals in the structure sheaf OX ′ of X ′ . Further, suppose that, given any affine open set U = Spec(AU ) contained in X ′ in which X is defin ...
... Examples. (a) Let X ′ be an integral noetherian scheme of finite type over C and let X be a closed integral subscheme of X ′ , corresponding to a sheaf I of ideals in the structure sheaf OX ′ of X ′ . Further, suppose that, given any affine open set U = Spec(AU ) contained in X ′ in which X is defin ...
Semicontinuous functions and convexity
... Proof. Suppose that f is lower semicontinuous and xα → x. Say t < f (x). Because f is lower semicontinuous, f −1 (t, ∞] ∈ τ . As x ∈ f −1 (t, ∞] and xα → x, there is some αt such that α ≥ αt implies xα ∈ f −1 (t, ∞]. That is, if α ≥ αt then f (xα ) > t. This implies that lim inf f (xα ) ≥ t. But th ...
... Proof. Suppose that f is lower semicontinuous and xα → x. Say t < f (x). Because f is lower semicontinuous, f −1 (t, ∞] ∈ τ . As x ∈ f −1 (t, ∞] and xα → x, there is some αt such that α ≥ αt implies xα ∈ f −1 (t, ∞]. That is, if α ≥ αt then f (xα ) > t. This implies that lim inf f (xα ) ≥ t. But th ...
SOME ASPECTS OF TOPOLOGICAL TRANSITIVITY——A SURVEY
... The two conditions are independent in general. In fact, take X = {0}∪{1/n : n ∈ N} endowed with the usual metric and f : X → X defined by f (0) = 0 and f (1/n) = 1/(n + 1), n = 1, 2, . . . . Clearly, f is continuous. The point x0 = 1 is (the only) transitive point for (X, f ) but the system is not t ...
... The two conditions are independent in general. In fact, take X = {0}∪{1/n : n ∈ N} endowed with the usual metric and f : X → X defined by f (0) = 0 and f (1/n) = 1/(n + 1), n = 1, 2, . . . . Clearly, f is continuous. The point x0 = 1 is (the only) transitive point for (X, f ) but the system is not t ...
An introduction to matrix groups and their applications
... which is a continuous function of the entries of A and so is a continuous function of A itself. Definition 1.14. Let G be a topological space and view G × G as the product space (i.e., give it the product topology). Suppose that G is also a group with multiplication map mult : G × G −→ G and inverse ...
... which is a continuous function of the entries of A and so is a continuous function of A itself. Definition 1.14. Let G be a topological space and view G × G as the product space (i.e., give it the product topology). Suppose that G is also a group with multiplication map mult : G × G −→ G and inverse ...
