Remove scare quotes. Change indexing to $n$-cells instead of $(n+1)$-cells

Author: GeoffreySangston

properties/P000240.md

@@ -12,12 +12,12 @@ refs:
 
 $X$ has a chain of subspaces $\empty = X_{-1}\subseteq X_0\subseteq X_1 \subseteq \dots$, such that:
 
-- $X_{n+1}$ can be obtained from $X_n$ by attaching $(n+1)$-cells.
-- $X = \bigcup_{n\geq 0}X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$.
+- $X_n$ is obtained from $X_{n-1}$ by attaching $n$-cells.
+- $X = \bigcup_{n\geq 0} X_n$ and a subspace $U \subseteq X$ is open iff $U \cap X_n$ open in $X_n$ for all $n \geq -1$.
 
-Here *attaching (n+1)-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^{n+1} \to X_n$, such that $X_{n+1}$ is homeomorphic to the quotient $(X_n \sqcup (J \times D^{n+1})) /{\sim}$,
-where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for all $(j,x)\in J \times \partial D^{n+1}$.
-The set $J$ is allowed to be empty, in which case $X_{n+1}=X_n$.
+Here *attaching n-cells* means that there is a discrete space $J$, and for every $j \in J$ a continuous map $f_j:\partial D^n \to X_{n-1}$, such that $X_n$ is homeomorphic to the quotient $(X_{n-1} \sqcup (J \times D^n)) /{\sim}$,
+where $\sim$ is the equivalence relation generated by $(j,x) \sim f_j(x)$ for all $(j,x)\in J \times \partial D^n$.
+The set $J$ is allowed to be empty, in which case $X_n=X_{n-1}$.
 The $0$-skeleton $X_0$ has the discrete topology; if it is empty, $X$ itself is empty.
 
 Here $D^n$ is the closed unit disk in $\mathbb R^n$ and $\partial D^n=S^{n-1}$ is the unit sphere in $\mathbb R^n$. We set $\partial D^0=S^{-1}=\emptyset$ by convention.