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\title{MATH 208 - HW \# 1}
\author{Nathan Marianovsky \\ October 4, 2017}

\begin{document}

\maketitle

\begin{problem}
Show that the set of points $(x,y) \in \mathbb{R}^2$ satisfying $xy = 0$ does not constitute a topological manifold.
\end{problem}

\vspace{.5cm}

\begin{solution}
We first note that all points satisfying the given relation belong to either the $x$-axis, $y$-axis, or both for the case of the origin. This means that at any point besides the origin we are locally homeomorphic to $\mathbb{R}$. We want to show it is impossible to find a homeomorphism that will take any neighborhood of the origin into $\mathbb{R}$ since it is homeomorphic to $\mathbb{R}^2$. Recall that connectedness is a property preserved by homeomorphisms. Thus, let $I_1,I_2,I_3$ and $I_4$ represent the four distinct intervals found in a neighborhood of the origin that intersect at a single point. Under a homeomorphism these intervals must remain distinct and intersect a single point. Unfortunately, this is impossible in $\mathbb{R}$ for more than two intervals because it would contradict the distinct assumption.
\end{solution}

\end{document}