1.1 Closed sets Let k be an algebraically closed field. Let A = Ak = {a = (a1, . . . , an) : ai ∈ k} = k be affine space. So A is a point, and A = k. Let An = k[T1, . . . , Tn] be the k-algebra of polynomials. If f ∈ An, a ∈ A, then f(a) ∈ k. Suppose S ⊆ An be a subset. Then Z(S) = {a ∈ A : f(a) = 0,∀f ∈ S} ⊆ A. A subset Z ⊆ A is called closed if Z = Z(S) for some S ⊆ An. Example 1.1. If n = 1,...

New and old results on closed polynomials, i.e., such polynomials f ∈ k[x1, . . . , xn]\k that the subalgebra k[f ] is integrally closed in k[x1, . . . , xn], are collected in the paper. Using some properties of closed polynomials we prove the following factorization theorem: Let f ∈ k[x1, . . . , xn] \ k, where k is algebraically closed. Then for all but finite number μ ∈ k the polynomial f + ...

For every Hausdorff space X the space XΘ is introduced. If X is H-closed, then XΘ is a quasy-compact T1-space. If f : X → Y is a mapping, then there exists the mapping fΘ : XΘ → YΘ. We say that a mapping f : X → Y is Θ-closed if fΘ is a closed mapping. If X and Y are H-closed and if f : X → Y is a HJ-mapping, then fΘ is Θ-closed. Let X = {Xa, pab, A} be an inverse system of H-closed spaces Xa a...

(4.1) Let E,F be nonempty subsets of R, E is compact and F is closed. Then there exist (e, f) ∈ E ×F such that d(E,F ) = d(e, f). proof Let α = d(E,F ) = inf(x,y)∈E×F d(x, y) be the distance between the two sets. Let > 0, and let x0 be a given element of E. Since E is compact, it is in particular bounded, and there exists r > 0 such that E ⊂ B(r, x0). Now consider the closed ball B(r + α+ , x0)...

On a Lebesgue measure space with measure element do, and total measure finite or infinite, we consider the complex-valued measurable functions f, y, so, ., each determined only a.e., and each belonging to all L,-classes simultaneously, 1<r< a. We donote (i) by F = {f} a ring of functions with complex constants and ordinary multiplication and closed under the involution: if f -= f + if2 e F then...

We show that if F is a smooth, closed, orientable surface embedded in a closed, orientable 3-manifold M such that for each Riemannian metric g on M , F is isotopic to a least-area surface F (g), then F is incompressible.

We show that for any closed surface F with χ(F ) −4 (or χ(F ) −2), there exist graphs that triangulate the torus or the Klein bottle (or the projective plane) and that quadrangulate F . We also give a sufficient condition for a graph triangulating a closed surface to quadrangulate some other surface. © 2006 Elsevier Inc. All rights reserved.

in an earlier work we showed that for ordered fields f not isomorphic to the reals r, there are continuous 1-1 unctions on [0, 1]f which map some interior point to a boundary point of the image (and so are not open). here we show that over closed bounded intervals in the rationals q as well as in all non-archimedean ordered fields of countable cofinality, there are uniformly continuous 1-1 func...

We show that there is some absolute constant c > 0, such that for any union-closed family F ⊆ 2, if |F| ≥ ( 1 2 − c)2n, then there is some element i ∈ [n] that appears in at least half of the sets of F . We also show that for any union-closed family F ⊆ 2, the number of sets which are not in F that cover a set in F is at most 2, and provide examples where the inequality is tight.

We show that the statement CCFC = “the character of a maximal free filter F of closed sets in a T1 space (X, T ) is not countable” is equivalent to the Countable Multiple Choice Axiom CMC and, the axiom of choice AC is equivalent to the statement CFE0 = “closed filters in a T0 space (X, T ) extend to maximal closed filters”. We also show that AC is equivalent to each of the assertions: “every c...