A selection theorem for simplex-valued maps

The purpose of this short note is to prove the following theorem. Let X be a hereditarily normal paracompact Hausdorff space, K be a simplicial complex, and : X K be a function. Suppose that {U | } and {f | } are collections such that for each , f is a map of U to |K|, and if x U, then f(x) (x). Assume further that {U | } is an open cover of X. Then there exists a map f : X |K| such that for each x X, f(x) (x)

A local to global selection theorem for simplex-valued functions

Suppose we are given a function : X K where X is a paracompact space and K is a simplicial complex, and an open cover {U | } of X, so that for each , f : U |K| is a map that is a selection of on its domain. We shall prove that there is a map f : X |K| which is a selection of . We shall also show that under certain conditions on such a set of maps or on the complex K, there exists a : X K with the property that each f is a selection of on its domain and that there is a selection f : X |K| of . The term selection, as used herein, will always refer to a map f, i.e., continuous function, having the property that f(x) (x) for each x in the domain

An addition theorem for n-fundamental dimension in metric compacta

AbstractWe generalize a notion of Freudenthal by proving that each metric compactum X is the inverse limit under an irreducible polyhedral representation of an extendable inverse sequence of compact triangulated polyhedra. The extendability criterion means that whenever X is a closed subspace of a metric compactum Y, then Y is the limit of an inverse sequence of polyhedra where all the bonding maps and triangulations are extensions of the one for X.We apply this to the theory of n-shape by using it to prove an addition theorem for n-fundamental (n-Fd) dimension. The theorem states that if a metric compactum Z is the union of two closed subspaces X1, X2 with X0 = X1 ∩ X2 and such that dim Z ⩽ n + 1, then n-Fd Z ⩽ max{n-Fd X1, n-Fd X2, n-Fd X0 + 1}

Approximate inverse limits and (m,n)-dimensions

In 2012, V. Fedorchuk, using m-pairs and n-partitions, introduced the notion of the (m,n)-dimension of a space. It generalizes covering dimension. Here we are going to look at this concept in the setting of approximate inverse systems of compact metric spaces. We give a characterization of (m,n)-dim X, where X is the limit of an approximate inverse system, strictly in terms of the given system

Čech systems and approximate inverse systems

We generalize a result of the first author who proved that the Čech system of open covers of a Hausdorff arc-like space cannot induce an approximate system of the nerves of these covers under any choices of the meshes and the projections

Simultaneous Z/p-acyclic resolutions of expanding sequences

We prove the following Theorem: Let X be a nonempty compact metrizable space, let $l_1 \leq l_2 \leq...$ be a sequence of natural numbers, and let $X_1 \subset X_2 \subset...$ be a sequence of nonempty closed subspaces of X such that for each k in N, $dim_{Z/p} X_k \leq l_k < \infty$. Then there exists a compact metrizable space Z, having closed subspaces $Z_1 \subset Z_2 \subset...$, and a surjective cell-like map $\pi: Z \to X$, such that for each k in N, (a) $dim Z_k \leq l_k$, (b) $\pi (Z_k) = X_k$, and (c) $\pi | {Z_k}: Z_k \to X_k$ is a Z/p-acyclic map. Moreover, there is a sequence $A_1 \subset A_2 \subset...$ of closed subspaces of Z, such that for each k, $dim A_k \leq l_k$, $\pi|{A_k}: A_k\to X$ is surjective, and for k in N, $Z_k\subset A_k$ and $\pi|{A_k}: A_k\to X$ is a UV^{l_k-1}-map. It is not required that X be the union of all X_k, nor that Z be the union of all Z_k. This result generalizes the Z/p-resolution theorem of A. Dranishnikov, and runs parallel to a similar theorem of S. Ageev, R. Jim\'enez, and L. Rubin, who studied the situation where the group was Z.Comment: 18 pages, title change in version 3, old title: "Z/p-acyclic resolutions in the strongly countable Z/p-dimensional case

Inverse systems of compact Hausdorff spaces and (m,n)-dimension

In 2012, V. Fedorchuk, using m-pairs and n-partitions, introduced the notion of the (m, n)-dimension of a space. It generalizes covering dimension; Fedorchuk showed that (m, n)-dimension is preserved in inverse limits of compact Hausdorff spaces. We separately have characterized those approximate inverse systems of compact metric spaces whose limits have a specified (m, n)-dimension. Our characterization is in terms of internal properties of the system. Here we are going to give a parallel internal characterization of those inverse systems of compact Hausdorff spaces whose limits have a specified (m, n)-dimension. Fedorchuk\u27s limit theorem will be a corollary to ours

Cell-like resolutions in the strongly countable Z-dimensional case

AbstractSuppose that X is a nonempty compact metrizable space and X1⊂X2⊂⋯ is a sequence of nonempty closed subspaces such that for each k∈N, dimZXk⩽k<∞. We show that there exists a compact metrizable space Z, having closed subspaces Z1⊂Z2⊂⋯, and a surjective cell-like map π:Z→X, such that for each k∈N, (a)dimZk⩽k,(b)π(Zk)=Xk, and(c)π|Zk:Zk→Xk is a cell-like map. Moreover, there is a sequence A0⊂A1⊂⋯ of closed subspaces of Z such that for each k, Zk⊂Ak, dimAk⩽k, π|Ak:Ak→X is surjective, and for k∈N, π|Ak:Ak→X is a UVk−1-map