I recently learned about the inductive dimension, but the formal definition is sometimes still a bit inaccessible. I have through experimentation however come upon my own definition for dimension which is easier for me to grasp, and I wonder if perhaps it aligns with either of the inductive dimensions:
Definition: A chain of balls $\mathcal{C}$ rooted in a Topological space $(X,\tau_X)$ is a sequence $$\partial B_1 \supseteq \partial B_2 ...$$ where each $B_{i+1}$ is the closure of a non-empty open set in $\partial B_i$ with the subspace Topology, and $B_1$ is the closure of an open set in $X$. The chain-dimension of $X$ is the supremum of the length of all such chains, if it exists, and $\infty$ otherwise.
Clearly the chain dimension of $\mathbb{R}$ is $2$, and $\mathbb{R}^2$ has chain dimension $3$, so for Euclidean spaces, it seems to be $1+$ the normal dimension. My question is if the same is true for either of the inductive dimensions?