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\begin{center}\begin{large}\textbf{Math 131 - Problem Set 7, Part 1\\
Due Tuesday, Oct 23}\end{large}\end{center}



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\noindent From Munkres: 51.2, 51.3


\begin{enumerate} 

\item Let $X$ be a topological space. Consider the equivalence relation on $X$ defined $x \sim y$ if there is a path in $X$ from $x$ to $y$. The equivalence classes are called \textit{path components} of $X$. Define $\pi_0(X)$ to be the set of path components of $X$.

\begin{enumerate} 

\item If $f : P \to Y$ is continuous and $P$ is path-connected, show that $f(P)$ is path-connected and thus contained in a single path component of $Y$. 
\item Show that if $f: X \to Y$ is a continuous function, there is an induced map on sets $\pi_0(f) : \pi_0(X) \to \pi_0(Y)$.
\item Show that $\pi_0$ is a functor from the category of topological spaces (with continuous functions) to the category of sets.

\end{enumerate}
\end{enumerate}

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