Decompositions of domains of proper local homeomorphisms

Topology and its Applications, 1984

A corollary of the main result of this paper is the following Theorem. Suppose f: X + Y is a closed sujection of metri:able spaces whose point inverses are LC"+'-divisors (n 2 1). If Y is complete and f is homology n-stable, then Y is LC"" provided X is LC"". Intuitively, f is homology n-stable if the Tech homology groups of its point inverses are locally constant up to dimension n. In addition, we obtain sufficient conditions for the Freudenthal compactification to be LC".

Publications de l'Institut Mathematique, 2011

Let X be a Hausdorff continuum (a compact connected Hausdorff space). Let 2 X (respectively, Cn(X)) denote the hyperspace of nonempty closed subsets of X (respectively, nonempty closed subsets of X with at most n components), with the Vietoris topology. We prove that if X is hereditarily indecomposable, Y is a Hausdorff continuum and 2 X (respectively Cn(X)) is homeomorphic to 2 Y (respectively, Cn(Y)), then X is homeomorphic to Y .

Algebraic & Geometric Topology

Wright [Wri92] showed that, if a 1-ended simply connected locally compact ANR Y with pro-monomorphic fundamental group at infinity (i.e. representable by an inverse sequence of monomorphisms) admits a Z-action by covering transformations, then that fundamental group at infinity can be represented by an inverse sequence of finitely generated free groups. Geoghegan and Guilbault [GG12] strengthened that result, proving that Y also satisfies the crucial semistability condition (i.e. representable by an inverse sequence of epimorphisms). Here we get a stronger theorem with weaker hypotheses. We drop the "promonomorphic hypothesis" and simply assume that the Z-action is generated by what we call a "coaxial" homeomorphism. In the pro-monomorphic case every Z-action by covering transformations is generated by a coaxial homeomorphism, but coaxials occur in far greater generality (often embedded in a cocompact action). When the generator is coaxial, we obtain the sharp conclusion: Y is proper 2-equivalent to the product of a locally finite tree with R. Even in the pro-monomorphic case this is new: it says that, from the viewpoint of fundamental group at infinity, the "end" of Y looks like the suspension of a totally disconnected compact set.

Journal of Mathematical Sciences & Computer Applications

In this paper, we first introduce a new class of closed map called   -closed map. Moreover, we introduce a new class of homeomorphism called   - Homeomorphism, which are weaker than homeomorphism. We also introduce  - Homeomorphisms and prove that the set of all   - Homeomorphisms form a group under the operation of composition of maps.2000 Math Subject Classification: 54C08, 54D05.

Topology and its Applications, 2008

A Bing space is a compact Hausdorff space whose every component is a hereditarily indecomposable continuum. We investigate spaces which are quotients of a Bing space by means of a map which is injective on components. We show that the class of such spaces does not include every compact space, but does properly include the class of compact metric spaces.

Proceedings of the American Mathematical Society, 1971

In this paper it is shown that for each positive integer n there is a locally compact Hausdorff space X having the property that dim X = n and in addition having the property that if f(X) = Y is a proper mapping, then dim Fa». Using this result it is shown that there is a space Y having the property that min dim Y = n with a point />E Y with min dim Y -\p\ =0.

We introduce new classes of sets called a-I-open, A- -I-open sets, A-pre-I-open sets, strongly T-I-sets, A- -T-I-sets, strongly BA-I-sets, BA-I-sets, and  A-I-open sets in ideal topological spaces. Using these sets, to obtain decompositions of continuity in an ideal topological spaces.

Let X be a locally 1-connected metric space and A 1 , A 2 , ..., A n be connected, locally path connected and compact pairwise disjoint subspaces of X. In this paper, we show that the quotient space X/(A 1 , A 2 , ..., A n ) obtained from X by collapsing each of the sets A i 's to a point, is also locally 1-connected. Moreover, we prove that the induced continuous homomorphism of quasitopological fundamental groups is surjective. Finally, we give some applications to find out some properties of the fundamental group of the quotient space X/(A 1 , A 2 , ..., A n ).

Topology and its Applications, 2006

We investigate the images (also called quotients) of countable connected bunches of arcs in R 3 , obtained by shrinking the arcs to points (see Section 2 for definitions of new terms). First, we give an intrinsic description of such images among T 1 -spaces: they are precisely countable and weakly first countable spaces. Moreover, an image is first countable if and only if it can be represented as a quotient of another bunch with its projection hereditarily quotient (Theorem 2.7). Applying this result we see, for instance, that two classical countable connected T 2 -spaces-the Bing space [R.H. Bing, A connected countable Hausdorff space, Proc. Amer. Math. Soc. 4 (1953) 474], and the Roy space [P. Roy, A countable connected Urysohn space with a dispersion point, Duke Math. J. 33 (1966) 331-333]-belong to such images. However, in these cases, we can show even more: each of the examples is a quotient, with hereditarily quotient projection, of a countable bunch of free segments (Examples 2.12 and 2.15). Next, we construct an example of a countable connected planar bunch of segments whose quotients are not first countable (Theorem 2.9). We also construct a collection of power c of countable connected Hausdorff spaces (with some extra properties). As a corollary we get that there exists a collection of power c of countable connected bunches of arcs in R 3 no two of which are homeomorphic (Theorem 3.1). We end this article with some open problems.

Topology and its Applications, 2012

a r t i c l e i n f o a b s t r a c t

American Journal of Mathematics, 1996

Let k be the k-dimensional Menger compact space. We show that the group Homeo (k) of homeomorphisms of k is acyclic. This result for the group of homeomorphisms with compact support of k fxg for a point x in k has been announced by Kawamura [7]. This theorem seems to be generalized for the group of homeomorphisms of a compact k-dimensional Menger manifold M. A compact k-dimensional Menger manifold is a compact Hausdorff space locally homeomorphic to the k-dimensional Menger compact space k. The homotopy groups of dimension less than k play an important role in the determination of compact k-dimensional

Proceedings of the American Mathematical Society, 1978

The existence (conjectured by R. Levy in a private communication) of a space X and an endohomeomorphism, /, of ßX, such that f[X] = ßX\X is demonstrated. It is shown that if G is one of the topological groups 2°, Q°, R° or T", where oi < a, then G has a dense C-embedded subgroup H and an autohomeomorphism, /, such that G is the union of disjoint sets, A0 and Ax, where for (i,j) = {0, 1} f[A¡\ = Ap and A i is a union of cosets of H. The existence (conjectured by R. Levy in a private communication) of a space X and an endohomeomorphism,/, of ßX, such that/[A'] = ßX\X is demonstrated. It is shown that if G is one of the topological groups 2a, Q", Ra or T°, where u < a, then G has a dense C-embedded subgroup H and an autohomeomorphism,/, such that G is the union of disjoint sets, A0 and Ax, where for (/',/} = {0, 1} f[A¡] = Aj, and A¡ is a union of cosets of H. These results, which continue (and duplicate in part) a remark of Glicksberg [G59], have appeared in [0'C76]. Notation. We denote the nonnegative integers, the first infinite ordinal and the corresponding countable discrete space by w. The topological groups 2, Z, Q, R and T are (0, 1}, the integers, the rationals, the reals and the circle group (R/Z), respectively. We denote the domain and the range of a function / by Dom / and Rng / respectively. Let / be a function, and let A he a subset of Dom/. An /-decomposition of A is a pair (A0, Ax} of complementary

2005

An ideal on a set X is a nonempty collection of subsets of X which sat- isfies the following conditions (1)A ∈ I and B ⊂ A implies B ∈ I; (2) A ∈ I and B ∈ I implies A ∪ B ∈ I. Given a topological space (X,  ) an ideal I on X and A ⊂ X, ℜa(A) is defined as ∪{U ∈  a : U−A ∈ I}, where the family of all a-open sets of X forms a topology [5, 6], denoted by  a. A topology, denoted  a  , finer than  a is generated by the basis (I,  ) = {V − I : V ∈  a(x), I ∈ I}, and a topology, denoted hℜa( )i coarser than  a is generated by the basis ℜa( ) = {ℜa(U) : U ∈  a}. In this paper A bijection f : (X, , I) → (X, , J) is called a A∗- homeomorphism if f : (X,  a  ) → (Y, a  ) is a homeomorphism, ℜa- homeomorphism if f : (X,ℜa( )) → (Y,ℜa()) is a homeomorphism. Properties preserved by A∗-homeomorphism are studied as well as nec- essary and sufficient conditions for a ℜa-homeomorphism to be a A∗- homeomorphism.

Assembling a localic map f : L → M from localic maps fi : Si → M , i ∈ J, defined on closed respectively open sublocales (J finite in the closed case) follows the same rules as in the classical case. The corresponding classical facts immediately follow from the behavior of preimages but for obvious reasons such a proof cannot be imitated in the point-free context. Instead, we present simple proofs based on categorical reasoning. There are some related aspects of localic preimages that are of interest, though. They are investigated in the second half of the paper.