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\centerline{\Large Math 675 Homework 4}
\centerline{Due 9/19/2018}
\vspace{.25in}

Responses must be typed, with a two-page maximum and LaTeX highly recommended. Grade will be based on style (2 points) and all of the problems.

\begin{enumerate}[1.]
\item Prove that the following are equivalent for a function $f: X\rightarrow Y$ between two metric spaces:
\begin{enumerate}[(i)]
\item $f$ is continuous,
\item $f^{-1}(A)$ is an open set for every open set $A\subset Y$,
\item $f^{-1}(B)$ is a closed set for every closed set $B\subset Y$.
\end{enumerate}
\item Prove that:
\begin{enumerate}[(a)]
\item The set of invertible 2-by-2 matrices is open. (Interpret the space of all 2-by-2 matrices as $\R^4$ with the usual topology.)
\item The set of determinant-one 2-by-2 matrices is closed.
\end{enumerate}
\item Let $A$ be a linearly independent set in a metric vector space $V$.
\begin{enumerate}[(a)]
\item Assuming that $V$ is finite-dimensional, prove that the set $\R A=\{\lambda a : \lambda \in \R, a\in A\}$ is closed.
\item Give an infinite-dimensional example for which $\R A$ is not closed.
\end{enumerate}
\item Provide examples of:
\begin{enumerate}[(a)]
\item An infinite union of closed sets that is not closed.
\item An infinite intersection of open sets that is not open.
\end{enumerate}
\end{enumerate}





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