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\lhead{\large \textbf{Math 461}}
\chead{\Large \textbf{Homework 4}}
\rhead{\large \textbf{DUE 9/30/2023}}
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{\Large\underline{\textbf{Exercises:}}} (\S $12$ and \S $13$)



\begin{enumerate}

\item Let $X$ be a topological space and let $A\subset X$ be a subset. Suppose that for all $x\in A$, there exists an open set $U$ satisfying $x\in U\subset A$. Show that $A$ is open.
	
\item Equip $\mathbb{R}$ with the standard topology. Show that a set $U\subset \mathbb{R}$ is open if and only if for all $x\in U$ there exists $\epsilon>0$ such that $(x-\epsilon, x+\epsilon)\subset U$.
	
\item Show that the collection $\mathcal{B}=\{(a,b)\colon a,b\in \Q\}$ is a basis for the standard topology on $\R$.  Conclude that standard topology on $\R$ therefore has a countable basis.
	
\item Let $X$ be a space.
	\begin{enumerate}
	\item Let $\{\mathcal{T}_i\mid i \in I\}$ be a non-empty collection topologies on $X$ (indexed by some set $I$). Show that $\bigcap_{i\in I} \mathcal{T}_i$ is a topology on $X$.
	
	\item Let $\mathcal{B}$ be a basis for a topology $\mathcal{T}$ on $X$. Show that $\mathcal{T}$ is the intersection of all topologies on $X$ that contain $\mathcal{B}$.
	
	\item Let $\mathcal{S}$ be a subbasis for a topology $\mathcal{T}$ on a space $X$. Suppose $\mathcal{T}'$ is another topology on $X$ that contains $\mathcal{S}$. Show that $\mathcal{T}$ is coarser than $\mathcal{T}'$.
	
	\item Let $\mathcal{S}$ and $\mathcal{T}$ be as in the previous part. Show that $\mathcal{T}$ is the intersection of all topologies on $X$ that contain $\mathcal{S}$.
	\end{enumerate}
	
	
\end{enumerate}




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