this PDF file - European Journal of Pure and Applied
... then it is β-open in the usual sense. Indeed, if A is I − β-open, then there is a preopen set G such that G \ A,and A \ cl(G) ∈ I = {;}, and so G ⊆ A ⊆ cl(G), proving that A is β-open. Conversely, suppose that whenever a set A is I − β-open, then it is β-open. Let B ∈ I. Then, B is I − β-open, and b ...
... then it is β-open in the usual sense. Indeed, if A is I − β-open, then there is a preopen set G such that G \ A,and A \ cl(G) ∈ I = {;}, and so G ⊆ A ⊆ cl(G), proving that A is β-open. Conversely, suppose that whenever a set A is I − β-open, then it is β-open. Let B ∈ I. Then, B is I − β-open, and b ...
Časopis pro pěstování matematiky - DML-CZ
... Definition 2.2. A function f : X -• yis said to be almost-continuous [6] if for each xeX and each open set Vcontaining f(x), there exists an open set U containing x such that f(U) c Inty(Clr(V)). It is obvious "that continuity implies strong semi-continuity and strong semicontinuity implies semi-co ...
... Definition 2.2. A function f : X -• yis said to be almost-continuous [6] if for each xeX and each open set Vcontaining f(x), there exists an open set U containing x such that f(U) c Inty(Clr(V)). It is obvious "that continuity implies strong semi-continuity and strong semicontinuity implies semi-co ...
local contractibility, cell-like maps, and dimension
... 1. Introduction. All spaces are separable metric. A compactum X has trivial shape if every continuous function from A' to a polyhedron is null homotopic. In addition, a continuous surjection /: X -» Y between compact spaces is called cell-like provided that f~l(y) has trivial shape for every y e Y. ...
... 1. Introduction. All spaces are separable metric. A compactum X has trivial shape if every continuous function from A' to a polyhedron is null homotopic. In addition, a continuous surjection /: X -» Y between compact spaces is called cell-like provided that f~l(y) has trivial shape for every y e Y. ...
Differential geometry for physicists
... A less strict definition would have been that of a topological atlas, where the transition functions only need to be continuous, or a C k -atlas for k ∈ N, where they need to be k times continuously differentiable. However, in physics it is often convenient to assume that everything is smooth, and s ...
... A less strict definition would have been that of a topological atlas, where the transition functions only need to be continuous, or a C k -atlas for k ∈ N, where they need to be k times continuously differentiable. However, in physics it is often convenient to assume that everything is smooth, and s ...
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.