Properties of faintly !-continuous functions 1. Introduction and
... Theorem 2.8. If f : (X, ⌧ ) ! (Y, ) is faintly !-continuous function and (Y, ) is almost-regular, then f is almost !-continuous. Proof. Let x 2 X and V be any regular open set of (Y, ) containing f (x). Since every regular open set in an almost-regular space is ✓-open [9], V is ✓open. Since f is fai ...
... Theorem 2.8. If f : (X, ⌧ ) ! (Y, ) is faintly !-continuous function and (Y, ) is almost-regular, then f is almost !-continuous. Proof. Let x 2 X and V be any regular open set of (Y, ) containing f (x). Since every regular open set in an almost-regular space is ✓-open [9], V is ✓open. Since f is fai ...
First-order difference equation
... The Sharkovskii ordering gives an ordering between all positive integers. ...
... The Sharkovskii ordering gives an ordering between all positive integers. ...
2 Continuous maps. 3 Induced topologies.
... 2 Continuous maps. Definition 2.1 Let (X, U) and (Y, V) a function f : X → Y is called continuous if the inverse image of any open set in Y is an open set in X. As a formula the definition is. ...
... 2 Continuous maps. Definition 2.1 Let (X, U) and (Y, V) a function f : X → Y is called continuous if the inverse image of any open set in Y is an open set in X. As a formula the definition is. ...
ON (g, s)-CONTINUOUS AND (πg, s)
... complement of π-open set is said to be π-closed. A subset A of a space X is said to be generalized closed (briefly, g-closed) [24] (resp. πg-closed [12]) if cl(A) ⊂ U whenever A ⊂ U and U is open (resp. π-open) in X. If the complement of A is πg-closed (resp. g-closed), A is said to be πg-open (resp ...
... complement of π-open set is said to be π-closed. A subset A of a space X is said to be generalized closed (briefly, g-closed) [24] (resp. πg-closed [12]) if cl(A) ⊂ U whenever A ⊂ U and U is open (resp. π-open) in X. If the complement of A is πg-closed (resp. g-closed), A is said to be πg-open (resp ...
paracompactness with respect to anideal
... U N b(Vs) c_ aP(Vs) Vs 6 2 (since 2" 7.), which comradiets the 7--boundary assumption of 2.. The following is an example of an Z-paracompact space (actually paraeompact) (X,7.), such that (X, (p(7.))) is not 2"-paracompact. EXAMPLE. Let X R with 7- the usual topology. Let 2" ((0,3)) {A _C XIA (0,3)} ...
... U N b(Vs) c_ aP(Vs) Vs 6 2 (since 2" 7.), which comradiets the 7--boundary assumption of 2.. The following is an example of an Z-paracompact space (actually paraeompact) (X,7.), such that (X, (p(7.))) is not 2"-paracompact. EXAMPLE. Let X R with 7- the usual topology. Let 2" ((0,3)) {A _C XIA (0,3)} ...
CHARACTERIZING CONTINUITY BY PRESERVING
... A and p 6∈ A for A ∈ A (because a component of G is Then p ∈ relatively closed in G), hence, by the definition of an s point, there exists a sequence {hxn , An i : n < ω} such that xn ∈ An ∈ A, xn → x for some x ∈ X and if An = Kn ∩ U then the components Kn are distinct. Note that x ∈ U ⊂ G. As dist ...
... A and p 6∈ A for A ∈ A (because a component of G is Then p ∈ relatively closed in G), hence, by the definition of an s point, there exists a sequence {hxn , An i : n < ω} such that xn ∈ An ∈ A, xn → x for some x ∈ X and if An = Kn ∩ U then the components Kn are distinct. Note that x ∈ U ⊂ G. As dist ...
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.