Functor of continuation in Hilbert cube and Hilbert space

TL;DR

This paper constructs a functor that extends maps between Z-sets in the Hilbert cube and Hilbert space to the entire space, with special properties, advancing the understanding of extension problems in infinite-dimensional topology.

Contribution

It introduces a functorial method for extending maps between Z-sets in the Hilbert cube and Hilbert space, with unique properties, enhancing extension theory in topology.

Findings

Existence of a functor extending maps between Z-sets to whole spaces.

The functor has special, proven properties.

Application to extension problems in infinite-dimensional topology.

Abstract

A $Z$-set in a metric space $X$ is a closed subset $K$ of $X$ such that each map of the Hilbert cube $Q$ into $X$ can uniformly be approximated by maps of $Q$ into $X \setminus K$. The aim of the paper is to show that there exists a functor of extension of maps between $Z$-sets of $Q$ [or $l_2$] to maps acting on the whole space $Q$ [resp. $l_2$]. Special properties of the functor are proved.