Z-Sets in large scale geometry

Abstract

This study introduces an analogous version of the Z-set in large-scale geometry, inspired by its foundational role in infinite-dimensional topology. A closed subset A ⊆ X is called Z-set of X, if there exist arbitrary small maps from X into X\A; that is, for every open cover U of X, there exists a map from X into X\A which is U-close to the identity. Although the Z-set does not seem very appealing, it is the most central concept in infinite-dimensional topology. Extending this idea to large-scale geometry, we define Coarse Z-sets by analyzing their behavior under arbitrarily small maps of X into X\A and examining their structural properties in a global context. A subset A ⊆ X is called Coarse Z-set if there exists a function from X into X\A that is “close” to identity map in the sense of large-scale geometry. Characterized by maps that are “close” to the identity, the Coarse Z-set can be thought of as a small set in a larger space; removing it does not change the overall structure of the original space. This study demonstrates that if a subset is a Coarse Z-set, the associated function is a quasi-isometry, guaranteeing coarse equivalence between the space and its complement. This equivalence preserves asymptotic dimensions, as expressed by asdim(X) = asdim(X \ A). Furthermore, Coarse Z- sets are invariant under coarse equivalence, showcasing their robustness in large-scale geometry.