Invariance of coarse z-sets under coarse embeddings

Abstract

The 𝑍-sets, as subsets of the Hilbert cube 𝑄, are a central concept in infinite-dimensional topology. In this study, we extend the notion of 𝑍-sets from infinite-dimensional topology to large-scale geometry, introducing the concept of Coarse 𝑍-sets. A closed subset 𝐴 ⊆ X is called a 𝑍-set of 𝑋 if for every open cover 𝑈 of 𝑋 and every function 𝑓 ∈ 𝐶(𝑄,𝑋) there exists a function 𝑔 ∈ 𝐶(𝑄,𝑋\𝐴) such that 𝑓 and 𝑔 are 𝑈-close. We define Coarse 𝑍-sets by analysing the classical definition of 𝑍-sets and examining their behavior under arbitrarily small maps from 𝑋 into 𝑋\𝐴, but now in a global, coarse geometric context. Specifically, a subset 𝐴 ⊆ X is called Coarse 𝑍-set if there exists a function from 𝑋 into X/𝐴 that is “close” to the identity map in the sense of large-scale geometry. In the current work, we aim to redefine the Coarse 𝑍-sets by using an analog of the Hilbert cube within large-scale geometry. We propose the Banach space 𝑙∞ as the analogue version of the Hilbert cube within large-scale geometry. While investigating this idea, we show that Coarse 𝑍-sets are invariant under coarse embeddings, if 𝑋 coarsely embeds into 𝑌, then the image of a Coarse 𝑍-set of 𝑋 under the embedding is a Coarse 𝑍-set of 𝑌. Consequently, the Coarse 𝑍-set of 𝑌 can be identified once the Coarse 𝑍-set of 𝑋 is known. As a result, this demonstrates the connection between Coarse 𝑍-set of a separable space and 𝑙∞ since any separable space 𝑋 can be coarsely embedded into the Banach space 𝑙∞. Thus, each Coarse 𝑍-set of a separable space 𝑋 can be mapped into 𝑙∞ due to the universality of 𝑙∞ for separable spaces.