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\usepackage{mathabx}




%% header, footer definitions
\pagestyle{fancy}
\fancyhead[L]{Math 441}
\fancyhead[C]{Homework 2} 
\fancyhead[R]{Due on July 3} 


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% theorem style
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\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
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\begin{document}

%\noindent Your name! \hfill  The date 

%\vskip10pt


\begin{problem}
Let $X$ be a set. Assume $\mathcal B$ and $\mathcal B'$ are bases for respective topologies on $X$. Prove that $\mathcal{T}(\mathcal{B}')$ is finer than  $\mathcal{T}(\mathcal{B})$ if and only if for each $B \in \mathcal{B}$ and each $x\in B$, there exists some $B' \in \mathcal{B}'$ such that $x\in B'\subseteq B$. 
\end{problem}

\begin{problem}
Consider on $\mathbb R$ the standard topology and the lower limit topology. Use the previous problem to decide which of these topologies is finer or coarser than the other.  
\end{problem}

\begin{problem}
Let $X$ be a set with more than one element. Prove that $\mathcal S=\{X\setminus \{x\} \mid x \in X\}$ is a subbasis for the cofinite topology on $X$. 
\end{problem}



\begin{problem}
Let $X$ and $Y$ be topological spaces with bases $\mathcal B$ and $\mathcal C$, respectively. Show that the collection 
\[\mathcal D=\{U \times V \subseteq X\times Y \mid U\in \mathcal B, V \in \mathcal C\}\]
is a basis for the product topology on $X \times Y$. 

\noindent \emph{Hint:} Lemma 13.2 from Munkres.
\end{problem}

\begin{problem}
Endow $\{0,1\}$ with the discrete topology. Let $\Lambda$ be a nonempty set. Prove that $ \prod_{i \in \Lambda} \{0,1\}$ (endowed with the product topology) is discrete if and only if $\Lambda$ is finite.
\end{problem}

\begin{problem}
Give a proof for Theorem 19.3 in Munkres.
\end{problem}

\end{document}