# On Closed Mappings of Sigma-Compact Spaces and Dimension

## TL;DR

This paper investigates the properties of closed images of certain sigma-compact spaces, showing they either contain sets with no transfinite dimension or arbitrarily high dimension, and constructs examples with dimension-preserving maps.

## Contribution

It provides new results on the dimension behavior of closed images of sigma-compact spaces, including specific constructions for dimension-preserving mappings.

## Key findings

- Non-one-point closed images of certain spaces contain sets with no transfinite dimension.
- Constructed sigma-compact spaces with images maintaining or exceeding original dimension.
- Examples for both finite and transfinite dimensions demonstrating dimension preservation.

## Abstract

We prove that if K is a remainder of the Hilbert space (i.e., K is the complement of the Hilbert space in its metrizable compactification) then every non-one-point closed image of K either contains a compact set with no transfinite dimension or contains compact sets of arbitrarily high inductive transfinite dimension ind. We construct also for each natural n a sigma-compact metrizable n-dimensional space whose image under any non-constant closed map has dimension at least n, and analogous examples for the transfinite dimension ind.