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\usepackage{mathtools}
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\usepackage{multicol}
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\usepackage{euler}
\usepackage[OT1]{eulervm}
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\usepackage{mathabx}




%% header, footer definitions
\pagestyle{fancy}
\fancyhead[L]{Math 441}
\fancyhead[C]{Homework 4} 
\fancyhead[R]{Due on July 17} 


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% theorem style
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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 $(X,d)$ be a metric space and let $\mathcal T_d$ denote the metric topology induced by $d$ and let $X_d$ denote the topological space $(X,\mathcal T_d)$.  
\begin{enumerate}
	\item Prove that $d\colon X_d \times X_d \to \mathbb R_{\text{std}}$ is continuous. 
	\item Prove that if $\mathcal T$ is another topology on $X$ such that $d\colon (X,\mathcal T) \times  (X,\mathcal T) \to \mathbb R_{\text{std}}$ is continuous, then $\mathcal T_d$ is coarser  than $\mathcal T$. 
\end{enumerate}
\textbf{Note:} This proves that the metric topology $\mathcal T_d$ is the coarsest topology on $X$ such that the given distance  function $d\colon X\times X \to \mathbb R$ is continuous. 
\end{problem}
\begin{problem}		
	
Let $(X,d)$ be a metric space. Define a new metric on $X$ by 
\[\overline{d}(x,y)=\frac{d(x,y)}{1+d(x,y)} \text{ for all }x,y\in X.\]
\begin{enumerate}
	\item Verify that $\overline{d}$ is a metric. 
	
	\emph{Hint:} For the triangle inequality of $\overline{d}$, use that of $d$ and the fact that the map $f\colon  [0,\infty) \to [ 0,1)$, where $f(x)=\frac{x}{1+x}$, is increasing and satisfies $f(a) + f(b) \geq f(a+b)$.
	
	\item Prove that $d$ and $\overline {d}$ induce the same topology.
	\emph{Hint:} Use Problem 1, and the fact that both $f$ and its inverse are continuous.
	
	\item For $X=\mathbb R$, show that $d$ and $\overline{d}$ are not equivalent in general.
\end{enumerate}
\end{problem}


\begin{problem}
Let $(X,d)$ be a metric space and let $Y\subseteq X$ be a subset. The restriction of $d$ to $Y$ defines a metric on $Y$
\[d_Y \colon Y\times Y \to [0,\infty), \quad d_Y(x,y)=d(x,y) \text{ for all }x,y\in Y.\]
Show that the metric topology on $Y$ associated to $d_Y$ coincides with the induced topology of $Y$ as a subspace of $X$. 
\end{problem}

\begin{problem}
Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. Assume that $f\colon X\to Y$ is a function such that 
\[d_Y(f(x_1), f(x_2))=d_X(x_1, x_2) \qquad \text{ for all } x_1, x_2 \in X\]
\begin{enumerate}
	\item Prove that $f$ is injective.
\item Prove that $f\colon X \to Y$ is continuous.
\item Prove that $f\colon X \to f(X)$ is open, where $f(X) \subseteq Y $ has the induced topology.
\end{enumerate}
\textbf{Note:} This proves that the corestriction $f\colon X \to f(X)$ is a homeomorphism. In this case, $f$ is said to be an imbedding of $X$ into $Y$.
\end{problem}

\begin{problem}
Let $X$ be a topological space and consider $X \times X$ with the product topology. Prove that $X$ is Hausdorff if and only if the \emph{diagonal} $\Delta =\{(x,y) \in X\times X \mid x=y\} $ is closed in $X\times X$.  
\end{problem}

\end{document}