Dimensional Compactum Research Articles

Locally contractible spaces that are absolute neighborhood retracts

It is shown that every locally contractible metric space that is the countable union of finite dimensional compacta is an absolute neighborhood retract.

On embeddings with locally nice cross-sections

A $k$-dimensional compactum ${X^k}$ in euclidean space ${E^n}(n - k \geqq 3)$ is said to be locally nice in ${E^n}$ if ${E^n} - {X^k}$ is $1$-ULC. In this paper we prove a general theorem which implies, in particular, that ${X^k}$ is locally nice in ${E^n}$ if the intersection of ${X^k}$ with each horizontal hyperplane of ${E^n}$ is locally nice in the hyperplane. From known results we obtain immediately that a $k$-dimensional polyhedron $P$ in ${E^n}$ ($n - k \geqq 3$ and $n \geqq 5$) is tame in ${E^n}$ if each $({E^{n - 1}} \times \{ w\} ) - P(w \in {E^1})$ is $1$-ULC. However, by strengthening our general theorem in the case $n = 4$, we are able to prove this result for $n = 4$ as well. For example, an arc $A$ in ${E^4}$ is tame if each horizontal cross-section of $A$ is tame in the cross-sectional hyperplane (that is, lies in an arc that is tame in the hyperplane).

Some Theorems on the Dimension of Fibre Spaces

THEoREM 1. Let a separcable metric space X be a fibre space over a potyhedron B with compact fibres. Then for any point b of B, I ) dim (b, B) + dim7rK1(b) dim (7r-1 (b), X). THEOREM 2. Let a finite dimensional compactum X be a fibre space over a polyhedron B. The subset B* of B which consists of all points b of B for which dim (b, B) + dim 7r(b) = dim (7r-1 (b), X) holds, contains an open dense subset of B.