Lecture notes (May 12)
... distinct points x, y ∈ X there are disjoint open sets U , V in X such that x ∈ U and y ∈ V . The indiscrete topology is manifestly not Hausdorff unless X is a singleton. The standard topology on Rn is Hausdorff: for x 6= y ∈ Rn , let d be half the Euclidean distance between x and y. Then U = Bd (x) ...
... distinct points x, y ∈ X there are disjoint open sets U , V in X such that x ∈ U and y ∈ V . The indiscrete topology is manifestly not Hausdorff unless X is a singleton. The standard topology on Rn is Hausdorff: for x 6= y ∈ Rn , let d be half the Euclidean distance between x and y. Then U = Bd (x) ...
Closure, Interior and Compactness in Ordinary Smooth Topological
... osct) on X or a gradation of closedness of ordinary subsets of X if C satisfies the following axioms : (OSCT1 ) C(∅) = C(X) = 1. (OSCT2 ) C(A ∪ B) ≥ C(A) ∧ C(B), ∀A, B ∈ 2X . ...
... osct) on X or a gradation of closedness of ordinary subsets of X if C satisfies the following axioms : (OSCT1 ) C(∅) = C(X) = 1. (OSCT2 ) C(A ∪ B) ≥ C(A) ∧ C(B), ∀A, B ∈ 2X . ...
The Proper Forcing Axiom and the Singular Cardinal Hypothesis
... and κ > ω1 is a regular cardinal with a nonreflecting stationary set consisting of points of countable cofinality, then κω1 = κ. This, combined with the above result of Veličković, strongly suggests that PFA implies SCH. In this paper we confirm this conjecture. The new technical tool we introduce ...
... and κ > ω1 is a regular cardinal with a nonreflecting stationary set consisting of points of countable cofinality, then κω1 = κ. This, combined with the above result of Veličković, strongly suggests that PFA implies SCH. In this paper we confirm this conjecture. The new technical tool we introduce ...
Spectra for commutative algebraists.
... K∗ (R) = π∗ (K(R)). Examples from geometric topology include the Whitehead space W h(X), Waldhausen’s K-theory of spaces A(X) [33] and the classifying space of the stable mapping class group BΓ+ ∞ [32]. We will give further details of some of these constructions later. 2.D. Fourth answer. This, fina ...
... K∗ (R) = π∗ (K(R)). Examples from geometric topology include the Whitehead space W h(X), Waldhausen’s K-theory of spaces A(X) [33] and the classifying space of the stable mapping class group BΓ+ ∞ [32]. We will give further details of some of these constructions later. 2.D. Fourth answer. This, fina ...
THE LINDEL ¨OF PROPERTY IN BANACH SPACES STUDIA MATH
... has d-diameter greater that ε. By induction on n = |s|, s ∈ 2(N) , we construct a family {Us : s ∈ 2(N) } of non-empty relatively τp -open subsets of C and a family {ts : s ∈ 2(N) } of points of D, satisfying the following conditions: (α) U∅ = C, τp τp (β) for each s, Us0 ∪ Us1 ⊂ Us , τp τp (γ) ρ(x( ...
... has d-diameter greater that ε. By induction on n = |s|, s ∈ 2(N) , we construct a family {Us : s ∈ 2(N) } of non-empty relatively τp -open subsets of C and a family {ts : s ∈ 2(N) } of points of D, satisfying the following conditions: (α) U∅ = C, τp τp (β) for each s, Us0 ∪ Us1 ⊂ Us , τp τp (γ) ρ(x( ...
