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\usepackage[OT1]{eulervm}
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\usepackage{mathabx}




%% header, footer definitions
\pagestyle{fancy}
\fancyhead[L]{Math 441}
\fancyhead[C]{Homework 6} 
\fancyhead[R]{Due on August 7} 


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% theorem style
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\newtheorem{corollary}[theorem]{Corollary}
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\newtheorem{defn}[theorem]{Definition}

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\newcommand*{\norm}[1]{\lvert \lvert #1\rvert\rvert}     %norm 

% operators
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\DeclareMathOperator*{\Var}{\mathbf{Var}}      % variance
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\begin{document}

%\noindent Your name! \hfill  The date 

%\vskip10pt

\begin{problem}
Let $X$ be a topological space and let $Y\subseteq X$ endowed with the induced topology. Prove that $Y$ is connected if and only if for every open sets $U$, $V$ of $X$ such that $Y\subseteq U\cup V$ and $Y\cap U \cap V= \emptyset$, we have either $Y\subseteq U$ or $Y \subseteq V$. 
\end{problem}

\begin{problem}
Determine when a set $X$ endowed with the cofinite topology is connected. Prove the resulting statement.
\end{problem}

\begin{problem}
Prove the following:

\begin{enumerate}
	\item A product of path-connected spaces is connected.
	\item If $f\colon X \to Y$ is continuous and $X$ is path-connected, then $f(X)$ is path-connected. 
	\item Any quotient of a path-connected space is path-connected.
\end{enumerate}
\end{problem}


\begin{problem} A subset $X \subseteq \R^n$ is called \emph{convex} if for every $x, y \in X$ and every $t \in [0, 1]$, we have  $tx + (1 - t)y\in X$ (in other words, the straight line segment from $x$ to $y$ is contained in $X$). Show that convex sets are path-connected. 
\end{problem}

\begin{problem}
Prove the following:
\begin{enumerate}
	\item For any $n>1$, the space $\R^n \setminus \{0\}$ is path connected. 
	\item For any $r\in \R$, the space $\R \setminus \{r\}$ is not connected. 
	\item For any $n>1$,  $\R^n$ and $\R$ are not homeomorphic. 
\end{enumerate}
\end{problem}

\begin{problem}
For $k\geq 1$, let  $S^k$ denote the unit sphere in $\R^{k+1}$, namely 
\[ S^k=\{(x_1, \dots, x_{k+1}) \in \R^{k+1} \mid \norm{x}= x_1^2 + \dots + x_{k+1}^2 =1 \}.\]
For every $n\geq 2$, prove that $\R^n\setminus \{0\}$ is homeomorphic to $S^{n-1}\times \R_{>0}$, where $\R_{>0}$ denotes the space of positive real numbers. You should write a formula for a homeomorphism and it's inverse, but you need not verify continuity.

Deduce that the sphere $S^n$ is path-connected for any $n\geq 1$.
\end{problem}

\begin{problem}
Let X be a topological space.
\begin{enumerate}
	\item Define a relation on $X$ by $x \sim y$ if and only if there exists a connected
	subspace $Y \subseteq X$ that contains both $x$ and $y$, for every $x, y \in X$. Verify
	that this is an equivalence relation. The equivalence classes are called the
	\emph{connected components} of $X$.
	
	\item  Define a relation on $X$ by $x \sim y$ if and only if there exists a path in $X$ from $x$ to $y$, for every $x, y \in X$. Verify
	that this is an equivalence relation. The equivalence classes are called the
	\emph{path components} of $X$.
\end{enumerate}
\end{problem}

\end{document}