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\begin{center} 
{\large {\bf Homework 8 }} \\
Topology, Math 441, Spring 2020 \\
Topic: Path-connectedness and Compactness \\
{\bf Due:} Wednesday April 29th, 2020 \\
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{\bf Reading:} pg 77-80, pg 83-89 the text book.    \\

{\bf Problems:}

\begin{enumerate}

\item Prove that the following spaces are path-commnected: \\
(a) $\R^2 - \{\overline{x}\}$ \hfill (b) $\R^n - \{\overline{x}\}$ is connected for $n \geq 3$,
\hfill (c) $S^{n-1} \subset \R^n$. 


\item Let $X=\{a,b\}$ and assume that $X$ has the topology 
$\mathcal{T}=\{\phi, \{a\}, X\}$. Prove that $X$ is path connected in this 
topology. 

\item Prove that the following spaces or subsets are non-compact.\\
(a) $(a,b) \subset \R$, \hfill (b) $[a,b) \subset \R$, \hfill (c) $\R^n$ \hfill \   

\item Prove that $X_{dis}$ is compact if and only if $X$ is finite. 

\item Prove that every subset $K \subset \R_{fc}$ is compact. 

\item Suppose that $X$ is a compact space and
  $(x_n)=\{x_1,x_2, \ldots \}$ is a sequence of points in $X$. Show
  that there is a subsequence of $(x_n)$ that converges to a point in
  $X$.

\item Prove that if $X \times Y$ is compact then so are $X$ and $Y$.
 
\item Prove that a union of two compacts subsets of $X$ is compact. 

\item[*9.] We have seen that the topologists sine curve is
  connected. Prove that it is not path-connected.

\item [*10.] Let $C$ be a countable subset of $\R^2$. Prove that 
the $A = \R^2 - C$ is path connected. (Hint: Given a point 
$\alpha \in A$, there are many lines through $\alpha$ that 
miss $C$ entirely. Why ? ). 



\end{enumerate}


{\bf Handin:} 1a, 3a, 4, 5, 8



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