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\title{Problem Set 9: The Fundamental Group}
\date{Due: Thursday, May 12}
\author{Your name:}





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\begin{comment}
\begin{prob}[8]
A space $X$ is said to be \emph{locally path connected at $x$} if for every neighborhood $U$ of $x$, there is a path connected neighborhood $V$ of $x$ contained in $U$. 
\begin{enumerate}
	\item Prove that path connected does \emph{not} imply locally path connected.
\end{enumerate}
\end{prob}
\end{comment}

\begin{prob}[8]
A \emph{based topological space} $(X, x_0)$ is a topological space $X$ together with a point $x_0 \in X$, called the basepoint. A morphism of based topological spaces from $(X, x_0)$ to $(Y,y_0)$ is a continuous map $f: X \to Y$ so that $f(x_0) = y_0$. Prove to yourself that this collection of objects and morphisms defines a category, which we denote $\text{Top}_*$.\\
\\
	\indent Prove that $\pi_1$ defines a functor $\text{Top}_* \to \text{Set}$.
\end{prob}

\begin{prob}[8]
Prove that $\pi_1(\D^n, 0) \cong \{\ast \}$.
\end{prob}

\begin{prob}[8]
Prove that for any point $x \in S^2$, $$\pi_1(S^2, x) \cong \{ \ast \}.$$ You may use without proof the fact that every map $(S^1, 1) \to (S^2,x)$ is homotopic (as a loop) to a map which is not surjective.\footnote{For those curious about the proof, you may wish to use the Lebesgue Number Lemma.}
\end{prob}

\begin{prob}[B16]
Prove that for any point $x \in \mathbb{RP}^2$, $\pi_1(\mathbb{RP}^2, x) \ncong \{\ast\}$. Conclude that $\mathbb{D}^2 \ncong \mathbb{RP}^2,$ $\mathbb{RP}^2 \ncong S^2$ (without any fancy theorems about curves separating the plane).
\end{prob}

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