Regular Cell Complex

Definition. Regular Cell Complex: A regular cell complex is a topological space $X$ partitioned into subspaces $\{X_\alpha\}_{\alpha\in PX}$ satisfying the following conditions: For each $x \in X$, every sufficiently small neighborhood of $x$ intersects finitely many $X_\alpha$. For all $\alpha, \beta$, $X_\alpha \cap X_\beta \neq \emptyset$ only if $X_\beta \subseteq X_\alpha$. Every $X_\alpha$ is homeomorphic to $\mathbb{R}^{n_\alpha}$ for some $n_\alpha$. For every $\alpha$, there is a homeomorphism of a closed ball in $\mathbb{R}^{n_\alpha}$ to $X_\alpha$ that maps the interior of the ball homeomorphically onto $X_\alpha$.