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\title{Homework 2}
\author{Due Monday, January 23, 2011}
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\begin{enumerate}

\item
\begin{enumerate}
\item Show that homotopy is an equivalence relation on the set of continuous maps $X \to Y$.
\item Suppose that $f_0, f_1: X \to Y$ and $g_0, g_1: Y \to Z$.  If $f_0 \simeq f_1$ and $g_0 \simeq g_1$, show that $g_0 \circ f_0 \simeq g_1 \circ f_1$.
\end{enumerate}

\item Let $f: S^1 \to X$.  Show that the following are equivalent:
\begin{enumerate}
\item $f$ is nullhomotopic.
\item $f$ extends to a map $D^2 \to X$---that is, there is a map $g: D^2 \to X$ such that the restriction $g|_{S^1} = f$.
\end{enumerate}

\item Let $X$ be a topological space.  Define $x \sim x'$ if there is a path from $x$ to $x'$.  Convince yourself that $\sim$ is an equivalence relation (but do not turn this in).  The equivalence classes of $\sim$ are called the \emph{path components} of $X$.  We define $\pi_0(X) = X/{\sim}$.
\begin{enumerate}
\item Show that a map $f: X \to Y$ induces a map $f_*: \pi_0(X) \to \pi_0(Y)$.
\item Show that homotopic maps $X \to Y$ induce the same map on $\pi_0$.
\item Show that a homotopy equivalence induces a bijection on $\pi_0$.
\end{enumerate}

\item What is one question you have about last week's lectures?

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