For a given ANR-sequence (X,A) associated with a par (X,A) of compacta, a pair (N(X),N(A)) of compact AR's containing (X,A) as an unstable pair is constructed. The weak proper homotopy type of the pair (N(X)-',N(A)-A) determines the shape of (X,A) in the sense of Mardešić and Segal. Several applications of this result are given. A cohomological version of the Whitehead theorem in shape theory is proved.
The product of images, under closed maps of metric spaces need not be a k-space. In view of these maps, we shall give some necessary conditions for products to be k-spaces.
For a locally compact space X we give a necessary and sufficient condition for every compactification aX of X with zero-dimensional remainder to be regular Wallman. As an application it follows that the Freudenthal compactification of a locally compact metrizable space is regular Wallman.
Let T: → be an (, )-topological functor and S: → a faithful functor. Let F: → and L: → be functors with a:FT → SL an epi natural transformation. We are concerned with the question of when L has a right adjoint given that F has a right adjoint. We give two characterizations of the existence of a right adjoint to L. One involves just the “topological data” and the other is an application of Freyd's adjoint functor theorem. As a consequence, we characterize when a category which is monoidal and (, )-topological over a monoidal closed category is also closed.
Let ƒ be a continuous map from a compact metric continuum X onto a continuum Y. Then ƒ is quasi-monotone if, for each subcontinuum K of Y with nonvoid interior, ƒ-1(K) has a finite number of components and each is mapped onto K by ƒ. Examples of quasi-monotone maps are local homeomorphisms and other finite to one confluent maps. In the following all maps are assumed to be quasi-monotone from X onto Y. A theorem of L. Mohier and J.B. Fugate [1] says that if X is irreducible between two of its points then Y is also irreducible between two of its points. This result is generalized to the following theorem. If X is irreducible about a finite point set A then either Y is irreducible about ƒ(A) or there is a point y in Y such that Y is irreducible about {y}⋃ƒ(A⧹{α}) for each a in A. Another result is that if X is a continuum that is separated by no subcontinuum, i.e., a θ1-continuum, then Y is a θ1-continuum or is irreducible between two of its points.
If there is a retraction from βX onto βX-X then X is locally compact and pseudocompact. (But X can have arbitrarily large closed discrete C∗-embedded subsets.)
This paper studies the relationships of θ-spaces to other generalized metric spaces. In particular, a condition shared by θ-spaces and spaces with a quasi-Gg diagonal is introduced, and it is shown that every regular θ refinable β-space satisfying this condition is semi-stratifiable. In addition, a regular quasi-complete space satisfying this condition has a base of countable order.
A complete survey of results on the epimorphism problem for various categories of topological groups is given. All known partial results on the epimorphism problem for Hausdorff topological groups support the conjecture that epimorphisms of Hausdorff topological groups have dense image.
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：