Coarse Structures on Infinite Groups
... (U1) if E ∈ E, then E −1 := {(y , x) ∈ X × X | (x, y ) ∈ E } ∈ E; (U2) if E , F ∈ E, then E ◦F := {(x, y ) ∈ X ×X | ∃z ∈ X s.t. (x, z) ∈ E , (z, y ) ∈ F }. (I1) and (I2) say that E is an ideal of X × X . Replacing (I1) and (I2) with their dual properties (F1) and (F2) (saying that E is a filter of X ...
... (U1) if E ∈ E, then E −1 := {(y , x) ∈ X × X | (x, y ) ∈ E } ∈ E; (U2) if E , F ∈ E, then E ◦F := {(x, y ) ∈ X ×X | ∃z ∈ X s.t. (x, z) ∈ E , (z, y ) ∈ F }. (I1) and (I2) say that E is an ideal of X × X . Replacing (I1) and (I2) with their dual properties (F1) and (F2) (saying that E is a filter of X ...
ro-PDF - University of Essex
... One of the first things one learns, as a student of measure theory, is that sets of measure zero are frequently ‘negligible’ in the straightforward sense that they can safely be ignored. This is not quite a universal principle, and one of my purposes in writing this treatise is to call attention to ...
... One of the first things one learns, as a student of measure theory, is that sets of measure zero are frequently ‘negligible’ in the straightforward sense that they can safely be ignored. This is not quite a universal principle, and one of my purposes in writing this treatise is to call attention to ...
pdf
... (2) The inverse image of each closed set in (Y, σ) is strongly G-βclosed; (3) For each x ∈ X and V ∈ σ containing f (x), there exists U ∈ SGβO(X) containing x such that f (U ) ⊆ V . Proof. The proof is obvious form Lemma 3.23 and is thus omitted. Theorem 4.3. A function f : (X, τ , G) → (Y, σ) is st ...
... (2) The inverse image of each closed set in (Y, σ) is strongly G-βclosed; (3) For each x ∈ X and V ∈ σ containing f (x), there exists U ∈ SGβO(X) containing x such that f (U ) ⊆ V . Proof. The proof is obvious form Lemma 3.23 and is thus omitted. Theorem 4.3. A function f : (X, τ , G) → (Y, σ) is st ...
FUZZY ORDERED SETS AND DUALITY FOR FINITE FUZZY
... for example, when expressing our preferences with a set of alternatives. Since then many notions and results from the theory of ordered sets have been extended to the fuzzy ordered sets. In [16], Venugopalan introduced a definition of fuzzy ordered set (foset) (P, µ) and presented an example on the ...
... for example, when expressing our preferences with a set of alternatives. Since then many notions and results from the theory of ordered sets have been extended to the fuzzy ordered sets. In [16], Venugopalan introduced a definition of fuzzy ordered set (foset) (P, µ) and presented an example on the ...
PDF version - University of Warwick
... Goresky and MacPherson have the additional condition p0 = p1 = p2 = 0 and King has no condition on p0 . However if pi > i then the intersection condition is vacuous, so we may as well assume p0 = 0. Geometry and Topology Monographs, Volume 2 (1999) ...
... Goresky and MacPherson have the additional condition p0 = p1 = p2 = 0 and King has no condition on p0 . However if pi > i then the intersection condition is vacuous, so we may as well assume p0 = 0. Geometry and Topology Monographs, Volume 2 (1999) ...
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.