Some Properties of θ-open Sets
... has the Euclidean topology and 2 = {0, 1} has the Serpiński topology with the singleton {0} open. Then A ⊂ X is θ-closed (θ-open, respectively) if and only if A = B × 2, where B ⊂ I is closed (open, respectively). Observe that if A ⊂ X is θ-closed, then Clθ (A) = A. Let B = πI (A) ⊂ I. Obviously, A ...
... has the Euclidean topology and 2 = {0, 1} has the Serpiński topology with the singleton {0} open. Then A ⊂ X is θ-closed (θ-open, respectively) if and only if A = B × 2, where B ⊂ I is closed (open, respectively). Observe that if A ⊂ X is θ-closed, then Clθ (A) = A. Let B = πI (A) ⊂ I. Obviously, A ...
Maximal Tychonoff Spaces and Normal Isolator Covers
... only if every continuous bijection from a space (Y, τ1 ) with the property P to (X, τ ) is a homeomorphism. In 1943 Hewitt [15] and in 1947 Vaidyanathaswamy [29] had independently proved that every compact Hausdorff space is maximal compact. Vaidyanathaswamy [29] put forward a question if there exis ...
... only if every continuous bijection from a space (Y, τ1 ) with the property P to (X, τ ) is a homeomorphism. In 1943 Hewitt [15] and in 1947 Vaidyanathaswamy [29] had independently proved that every compact Hausdorff space is maximal compact. Vaidyanathaswamy [29] put forward a question if there exis ...
Algebraic characterization of finite (branched) coverings
... Our starting point is the well-known result which states that, in the realm of realcompact spaces, every space X is determined by the algebra C(X) of all real-valued continuous functions defined on it, and that continuous maps between such spaces are in one-to-one correspondence with homomorphisms b ...
... Our starting point is the well-known result which states that, in the realm of realcompact spaces, every space X is determined by the algebra C(X) of all real-valued continuous functions defined on it, and that continuous maps between such spaces are in one-to-one correspondence with homomorphisms b ...
ANALOGUES OF THE COMPACT-OPEN TOPOLOGY M. Schroder
... Q . A subset K of Q is said to be compact (or more precisely, y-compact) if every ultra-filter on Q to which K belongs y-converges to some point of K , and y itself is called compact if Q is y-compact. ...
... Q . A subset K of Q is said to be compact (or more precisely, y-compact) if every ultra-filter on Q to which K belongs y-converges to some point of K , and y itself is called compact if Q is y-compact. ...
MAT1360: Complex Manifolds and Hermitian Differential Geometry
... A “complex manifold” is a smooth manifold, locally modelled on the complex Euclidean space Cn and whose transition functions are holomorphic. More precisely, a complex manifold is a pair (M, J) consisting of a smooth, real manifold of real dimension 2n and a maximal atlas whose overlap maps lie in t ...
... A “complex manifold” is a smooth manifold, locally modelled on the complex Euclidean space Cn and whose transition functions are holomorphic. More precisely, a complex manifold is a pair (M, J) consisting of a smooth, real manifold of real dimension 2n and a maximal atlas whose overlap maps lie in t ...
Generalities About Sheaves - Lehrstuhl B für Mathematik
... Definition Let {Vi } be an open covering of U ⊆ X (open). A presheaf F on X is a sheaf if for all i: s ∈ F(U ) and s|Vi = 0 then s = 0, given si ∈ F(Vi ) that match on the overlaps: si |Vi ∩Vj = sj |Vi ∩Vj there is a unique s ∈ F(U ) with s|Vi = si . Sheaves are defined by local data. ...
... Definition Let {Vi } be an open covering of U ⊆ X (open). A presheaf F on X is a sheaf if for all i: s ∈ F(U ) and s|Vi = 0 then s = 0, given si ∈ F(Vi ) that match on the overlaps: si |Vi ∩Vj = sj |Vi ∩Vj there is a unique s ∈ F(U ) with s|Vi = si . Sheaves are defined by local data. ...
NON-ARCHIMEDEAN ANALYTIFICATION OF ALGEBRAIC
... Second, with no local separatedness assumptions on U it does not suffice to assume that δ is compact. In fact, in Example 5.1.4 we give examples of (non-separated) compact Hausdorff U and a free right action on U by a finite group G such that U/G does not exist. In such cases the action map δ : R = ...
... Second, with no local separatedness assumptions on U it does not suffice to assume that δ is compact. In fact, in Example 5.1.4 we give examples of (non-separated) compact Hausdorff U and a free right action on U by a finite group G such that U/G does not exist. In such cases the action map δ : R = ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.