spaces every quotient of which is metrizable
... [1, 2, 6, 7]. In an elementary course in general topology we learn that every continuous image in Hausdorff space of a compact metric space is metrizable [8]. In [7], Willard proved that every closed continuous image of a metric space X in Hausdorff space is metrizable if and only if the set of all ...
... [1, 2, 6, 7]. In an elementary course in general topology we learn that every continuous image in Hausdorff space of a compact metric space is metrizable [8]. In [7], Willard proved that every closed continuous image of a metric space X in Hausdorff space is metrizable if and only if the set of all ...
A note on the precompactness of weakly almost periodic groups
... If the answer is in the armative, then our Main Theorem 4.5 below follows at once, and Proposition 4.3 and Lemma 4.4 are not needed. We can only provide a positive answer to Question 3.5 under the additional assumption that G is ! bounded. A topological group is called ! -bounded if for every U 2 N ...
... If the answer is in the armative, then our Main Theorem 4.5 below follows at once, and Proposition 4.3 and Lemma 4.4 are not needed. We can only provide a positive answer to Question 3.5 under the additional assumption that G is ! bounded. A topological group is called ! -bounded if for every U 2 N ...
Section I. TOPOLOGICAL SPACES
... Definition 4С.12. The topological space is Hausdorf space if all its different points have neighbourhoods with empty intersection. The space is Hausdorf one iff each its sequence can have lass or equal one limit. For example, only (X,1) from the two point’s spaces is Hausdorf space. The points х an ...
... Definition 4С.12. The topological space is Hausdorf space if all its different points have neighbourhoods with empty intersection. The space is Hausdorf one iff each its sequence can have lass or equal one limit. For example, only (X,1) from the two point’s spaces is Hausdorf space. The points х an ...
279 ASCOLI`S THEOREM IN ALMOST QUIET QUASI
... Definition 1.6. [7] A set A ⊂ (X, τ ) is said to be N-closed in X or simply N-closed, if for any cover of A by τ -open sets, there exists a finite subcollection the interiors of the closures of which cover A; interiors and closures are of course w.r.t. τ . A set (X, τ ) is said to be nearly compact ...
... Definition 1.6. [7] A set A ⊂ (X, τ ) is said to be N-closed in X or simply N-closed, if for any cover of A by τ -open sets, there exists a finite subcollection the interiors of the closures of which cover A; interiors and closures are of course w.r.t. τ . A set (X, τ ) is said to be nearly compact ...
Topology Proceedings 7 (1982) pp. 293
... nEw, a E wI' let fn(a) < a be such that, for some B E wI' the B,fn(a)th neighborhood of (a,a,n) is contained in V. Each f n is a regressive function on wI and so there is an uncountable set An contained in wI and An E wI such that, ...
... nEw, a E wI' let fn(a) < a be such that, for some B E wI' the B,fn(a)th neighborhood of (a,a,n) is contained in V. Each f n is a regressive function on wI and so there is an uncountable set An contained in wI and An E wI such that, ...
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.