fix: proofread Algebraic Topology I: Homotopy part

tex/topology/constructions.tex

@@ -697,7 +697,7 @@ \subsection{Complex projective space}
 \end{remark}
 
 Like $\RP^n$, $\CP^n$ is a CW complex, built inductively
-by taking $\CC^n$ and welding its boundary onto $\CP^{n-1}$
+by taking $\CC^n$ and welding its boundary onto $\CP^{n-1}$.
 The difference is that as topological spaces,
 \[ \CC^n \cong \RR^{2n} \cong D^{2n}. \]
 Thus, we attach the cells $D^0$, $D^2$, $D^4$ and so on

tex/topology/cover-project.tex

@@ -357,7 +357,7 @@ \section{Lifting theorem}
 			draw(sine_curve -- loop, linewidth(1.8) + roundjoin);
 		\end{asy}
     \end{center}
-    The space $Y$ is simply connected, thus the lifting criteria
+    The space $Y$ is simply connected, thus the lifting criterion
 	is trivially satisfied, but the function $f:Y\to S^1$ which
 	collapses the sine curve vertically does not lift to a function
 	$\tilde f:Y\to\RR$. The lifting theorem does not apply here,
@@ -439,7 +439,7 @@ \section{Lifting correspondence}
 	You can show (and it's intuitively obvious) that the bijection
 	\[ \Phi \colon \pi_1(S^1) \leftrightarrow \ZZ \]
 	is in fact a group homomorphism if we equip $\ZZ$ with its
-	additive group structure $\ZZ$.
+	additive group structure.
 	Since it's a bijection, this leads us to conclude $\pi_1(S^1) \cong \ZZ$.
 \end{example}
 
@@ -566,7 +566,7 @@ \section{The algebra of fundamental groups}
 
 Let $X$ and $Y$ be topological spaces and $f \colon (X, x_0) \to (Y, y_0)$.
 Recall that we defined a group homomorphism
-\[ f_\sharp \colon \pi_1(X, x_0) \to \pi_1(Y_0, y_0)
+\[ f_\sharp \colon \pi_1(X, x_0) \to \pi_1(Y, y_0)
 	\quad\text{by}\quad
 	[\gamma] \mapsto [f \circ \gamma]. \]
 % which gave us a functor $\catname{Top}_\ast \to \catname{Grp}$.

tex/topology/fundamental-group.tex

@@ -231,7 +231,7 @@ \section{Fundamental groups}
 		that it winds around the origin -- these are so-called
 		\emph{winding numbers}.  Think about it!
 		\ii Similarly, we will soon show that the fundamental group of $S^1$
-		(the boundary of the unit circle) is $\ZZ$.
+		(the unit circle) is $\ZZ$.
 	\end{enumerate}
 	Officially, I also have to tell you what the base point is, but
 	by symmetry in these examples, it doesn't matter.
@@ -264,7 +264,7 @@ \section{Fundamental groups}
 	is the \emph{free group} generated on two letters.
 	The idea is that one loop of the eight is $a$,
 	and the other loop is $b$, so we expect $\pi_1$
-	to be generated by this loop $a$ and $b$ (and its inverses
+	to be generated by these loops $a$ and $b$ (and their inverses
 	$\ol a$ and $\ol b$).
 	These loops don't talk to each other.
 	\begin{center}
@@ -630,7 +630,7 @@ \section{The pointed homotopy category}
 	\end{tikzcd}
 	\end{center}
 \end{theorem}
-The fact that $\pi_1$ is a functor instead of merely assigns some group $\pi_1(X, x_0)$
+The fact that $\pi_1$ is a functor instead of merely assigning some group $\pi_1(X, x_0)$
 to each pointed topological space $(X, x_0)$ automatically implies several nice things, like:
 \begin{itemize}
 	\ii The functor bundles the information of $f_\sharp$,