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\noindent 
{\sc ºÚÁÏÍø Topology
Comprehensive Exam,  Feb. 15, 2025.} 

\noindent\makebox[6in]{\hrulefill} 

Answer six (6) questions total. On each page of your answers, please 
do not write anything on the back of that page.

\noindent\makebox[6in]{\hrulefill} 

\noindent {\bf 1}).
Determine whether the following statements about {\em closure} are true. 
If a statement is true, 
please give a proof. If a statement is not 
true, give a counter example.

i. $\overline{A_1}\cup \overline{A_2}\cup\overline{A_3}\cup ... = 
\overline{A_1\cup A_2\cup A_3 ...}$ 

ii. $\overline{A-B}=\overline{A}-\overline{B}$. 

iii. $\overline{A\cap B}=\overline{A}\cap\overline{B}$. 

\vspace{5mm}

\noindent {\bf 2}). 
i. Assume $X=\bigcup_{k=1}^\infty W_k$, where  
$W_k$ is open for any $k$. Assume $f:X\to Y$ 
is a map, so that $f|_{W_k}$ (i.e. the restriction of $f$ on $W_k$, 
viewed as a map $W_k\to Y$) is continuous for any $k$. Prove that $f$ 
is continuous. 

ii. 
Assume $X=A\cup B$, where  
$A, B$ are closed. Assume $f: X\to Y$ 
is a map, so that $f|_A$ and $f|_B$ are continuous. Prove that $f$ 
is continuous. 

iii. 
Assume $X=\bigcup_{k=1}^\infty E_k$, where  
$E_k$ is closed for any $k$. Assume $f:X\to Y$ 
is a map, so that $f|_{E_k}$ is continuous for any $k$. Is it always 
true that $f$ is continuous? (Prove it or give a counterexample.)


\vspace{5mm}

\noindent {\bf 3)}. 
i.  Prove that if $X$ is connected, $f : X\rightarrow Y$ is continuous, 
then $f(X)$ is connected. 

ii. If the product space $\prod_{\alpha\in A}X_\alpha$ 
contains a nonempty connected open set $U$,  
prove that $X_\alpha$ is connected for all but finitely many $\alpha$.

\vspace{5mm}

\noindent {\bf 4)}. 
Let $\tau$ be the smallest topology on ${\mathbb R}^2$ such that the intersection of any two lines is open.  

i. Is $({\mathbb R}^2,\tau)$ 1st countable?

ii. Is $({\mathbb R}^2,\tau)$ 2nd countable?

iii. Is $({\mathbb R}^2,\tau)$ metrizable?

Give your reasons. 


\vspace{5mm}

\noindent 
{\bf 5)}. Define an equivalence relation $\sim$ on ${\mathbb R}^2$: 
$(x, y)\sim (a, b)$ if and only if 
\[
3x-5y=3a-5b.
\] 

Prove that this is an equivalence relation, and the quotient 
space ${\mathbb R}^2/\sim$ is homeomorphic to ${\mathbb R}$. 

\clearpage

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\noindent {\bf 6}). 
i. Assume $X$ is a metric space with distance $d$. For a subset 
$S\subset X$, define the function $d_S$ by 
\[
d_S(x)=\inf_{q\in S} d(x, q). 
\]  
Prove that $f$ is continuous. 

ii. Prove that any metric space is normal. 


\vspace{1cm}




\noindent {\bf 7}). Assume $X_1, X_2, X_3, ...$ is a sequence of topological spaces, 
$Y_1, Y_2, Y_3,  ...$ is a sequence of topological spaces. Assume for 
each $j=1, 2, 3, ...$, there is a map $f_j: X_j\rightarrow Y_j$. Define
\[
f: \prod_{j=1}^\infty X_j \longrightarrow \prod_{j=1}^\infty Y_j
\]
where both domain and range have the product topology, by 
\[
f(x_1, x_2, x_3, ... ...)=\big(f_1(x_1), f_2(x_2), f_3(x_3), ...  ...\big).
\]
Prove that $f$ is continuous if and only if every $f_j$ is continuous. 


\vspace{1cm}



\noindent {\bf 8}). 
i. Give an example of a topological space $X$, and a sequence of nonempty 
closed subsets $A_1, A_2, ... \subset X$, so 
that $A_1\supset A_2 \supset A_3\supset ...$ and 
$\bigcap_{n=1}^\infty A_n=\emptyset$.

ii. Assume $X$ is compact, $A_1\supset A_2 \supset A_3\supset ...$ 
is a sequence of 
closed nonempty subsets. Prove that  $\bigcap_{n=1}^\infty A_n$ is not empty. 

\vspace{1cm}


\noindent {\bf 9)}. 
%i.  Prove that a compact subset of a Hausdorff space is closed.
i. Prove that a closed subset of a compact space is compact.

ii. Prove that in a Hausdorff space, for any compact set $K$ and a point 
$p\notin K$, there are disjoint open sets $U, V$ such that $K\subset U$ and 
$p\subset V$. 

\vspace{1cm}


\noindent {\bf 10}). 
Consider the product space $X=\prod_{n=1}^\infty [0,1]$, with 
the product topology. 

i. Prove that $X$ is Hausdorff. 

ii. Prove that $X$ is separable. 



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