👉 The covering ion, also known as the covering function or the covering number, is a concept in topology that measures the minimum number of open sets needed to cover a topological space. Formally, given a topological space \( X \) and an open cover \( \{U_i\}_{i \in I} \), the covering ion \( \kappa(X) \) is the smallest integer such that \( X \) can be covered by \( 2^{\kappa(X)} \) (the set of all subsets of \( X \) with cardinality at most \( \kappa(X) \)) open sets. This function provides a way to quantify how "dense" or "spread out" the open cover is, essentially giving an upper bound on the number of sets required to fully cover the space. It's a crucial tool in understanding the topological properties of spaces and plays a significant role in areas like combinatorial topology and the study of manifolds.
