“Quasi-uniform spaces”
... d on a set X is a function d : X × X → [0, ∞) that satisfies d(x, x) = 0 and d(x, z) ≤ d(x, y)+d(y, z) whenever x, y, z ∈ X. Then d−1 defined by d−1 (x, y) = d(y, x) whenever x, y ∈ X is the conjugate quasi-pseudometric of d. A quasipseudometric d on X is called a quasi-metric if x, y ∈ X and d(x, ...
... d on a set X is a function d : X × X → [0, ∞) that satisfies d(x, x) = 0 and d(x, z) ≤ d(x, y)+d(y, z) whenever x, y, z ∈ X. Then d−1 defined by d−1 (x, y) = d(y, x) whenever x, y ∈ X is the conjugate quasi-pseudometric of d. A quasipseudometric d on X is called a quasi-metric if x, y ∈ X and d(x, ...
NOTES ON THE STRUCTURE OF SOLUTION SPACES Throughout
... with A(t) being an n × n matrix of functions and b(t) being an ncolumn vector of function. Here A(t), b(t) are continuous. Part I. The structure of the solution space to the homogeneous case. a. The solution space to T v = 0 forms a subspace, Ker(T ), of V . b. The solution space to Ax = 0 forms a s ...
... with A(t) being an n × n matrix of functions and b(t) being an ncolumn vector of function. Here A(t), b(t) are continuous. Part I. The structure of the solution space to the homogeneous case. a. The solution space to T v = 0 forms a subspace, Ker(T ), of V . b. The solution space to Ax = 0 forms a s ...
The computer screen: a rectangle with a finite number of points
... theorem is the key tool. Recall that a Jordan curve is a homeomorphic (= continuous one-one, inverse continuous) image of the circle; equivalently, it is a continuous image of [0, 1] under a map which is one-one on [0, 1) and f (0) = f (1). Then: Jordan curve theorem. If a Jordan curve, J, is remove ...
... theorem is the key tool. Recall that a Jordan curve is a homeomorphic (= continuous one-one, inverse continuous) image of the circle; equivalently, it is a continuous image of [0, 1] under a map which is one-one on [0, 1) and f (0) = f (1). Then: Jordan curve theorem. If a Jordan curve, J, is remove ...
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.