We define the dimension of X at p to be \[\dim_p X \overset{\text{def}}{=} \max \bigl\{ k \in \mathbb{N}_0 : \text{ $\forall$ neighbhds. $W$ of $p$, $\exists \, q \in W \cap X_{\mathit{reg}}$ ...We define the dimension of X at p to be dimpXdef=max{k∈N0:∀ neighbhds. W of p, ∃q∈W∩Xreg with dimqX=k}. If (X,p) is a germ and X a representative, the dimension of (X,p) is the dimension of X at p.
