6 | Continuous Functions
... 6.12 Note. If � : X → Y is a continuous bijection then � need not be a homeomorphism since the inverse function � −1 may be not continuous. For example, let X = {�1 � �2 } be a space with the discrete topology and let Y = {�1 � �2 } be a space with the antidiscrete topology. Let � : X → Y be given b ...
... 6.12 Note. If � : X → Y is a continuous bijection then � need not be a homeomorphism since the inverse function � −1 may be not continuous. For example, let X = {�1 � �2 } be a space with the discrete topology and let Y = {�1 � �2 } be a space with the antidiscrete topology. Let � : X → Y be given b ...
An up-spectral space need not be A
... C = U ∪ (∪{Ut ∩ (X ω \ Vt ) : t ∈ T }) , where U, Ut , Vt ∈ Ω(X). Clearly, C is compact. Now, one can prove that the set C is co-ICO by induction (on the cardinality n of T ), using Remark 1.2 and the equality W ∪ (Un+1 ∩ (X \ Vn+1 )) = (W ∪ Un+1 ) ∩ ...
... C = U ∪ (∪{Ut ∩ (X ω \ Vt ) : t ∈ T }) , where U, Ut , Vt ∈ Ω(X). Clearly, C is compact. Now, one can prove that the set C is co-ICO by induction (on the cardinality n of T ), using Remark 1.2 and the equality W ∪ (Un+1 ∩ (X \ Vn+1 )) = (W ∪ Un+1 ) ∩ ...
An up-spectral space need not be A-spectral
... C = U ∪ (∪{Ut ∩ (X ω \ Vt ) : t ∈ T }) , where U, Ut , Vt ∈ Ω(X). Clearly, C is compact. Now, one can prove that the set C is co-ICO by induction (on the cardinality n of T ), using Remark 1.2 and the equality W ∪ (Un+1 ∩ (X \ Vn+1 )) = (W ∪ Un+1 ) ∩ ...
... C = U ∪ (∪{Ut ∩ (X ω \ Vt ) : t ∈ T }) , where U, Ut , Vt ∈ Ω(X). Clearly, C is compact. Now, one can prove that the set C is co-ICO by induction (on the cardinality n of T ), using Remark 1.2 and the equality W ∪ (Un+1 ∩ (X \ Vn+1 )) = (W ∪ Un+1 ) ∩ ...
Metric and Banach spaces
... Theorem B.2 Let (X, dX ) and (Y, dY ) be two metric spaces and let consider a uniformely continuous function f : (X, dX ) → (Y, dY ). If (xn )n∈N is a Cauchy sequence of X, then f (xn )n∈N is a Cauchy sequence of F . The reciprocal one is not true. Proposition B.6 We have two properties about conver ...
... Theorem B.2 Let (X, dX ) and (Y, dY ) be two metric spaces and let consider a uniformely continuous function f : (X, dX ) → (Y, dY ). If (xn )n∈N is a Cauchy sequence of X, then f (xn )n∈N is a Cauchy sequence of F . The reciprocal one is not true. Proposition B.6 We have two properties about conver ...
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.