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\begin{center}
{\Large Intro to Topology -- Hw2.}
\end{center}
\bigskip\noindent
{\bf Always give reasons.}

\begin{enumerate}

\item Consider a metric space $(X,d)$ (that is a set $X$ with a metric $d$ on it)
and fix $p\in X$ as well as
$r>0$. Define a closed ball centered at $p$, radius $r$ as
$$\overline{B_p(r)}=\{ q\in X\,|\, d(p,q)\leq r \}$$

Show that a closed ball is a closed set.

\item 
a.) 
For an infinite set $X$ consider the collection of subsets
$$\tau=\{\emptyset,X\}\cup\{ U\subset X\,|\, X\setminus U\mbox{ is finite
}\}$$.


Show that $\tau$ is a topology on $X$. (It is called the {\it
co-finite} topology on $X$.)

\noindent b.) 
Let $X=\N$ and consider $g:\N\to\N$  given by

$g(x)=\left\{ \begin{array}{ll}
                2 &\mbox{ if } x \mbox{ is even }\\
                4 &\mbox{ if } x \mbox{ is odd }
                
               \end{array}
               \right .
               $

If $\N$ is equipped with the co-finite topology both as a source
(domain) and target (range), is $g$ continuous?


\item Let $f:X\to Y$ be a mapping between topological spaces $(X,\tau_X)$ and $(Y,\tau_Y)$. Prove that $f$ is continuous if and only if $\forall V\subset Y$
such that $V$ is closed, we have $f^{-1}(V)$ is closed in $X$.

\item Let $f:X\to Y$ be a mapping between topological spaces $(X,\tau_X)$ and $(Y,\tau_Y)$. 
For each of the types of functions I.)  and II.) below decide if the given properties are equivalent to $f$ being continuous. If yes, prove it.

For the one(s) not equivalent, decide if continuity implies the given property or vice-versa. If one implies the other, prove it. If not, provide an example showing it does not.

Property I.) $f$ takes open sets to open sets. (Such a map is called an {\it open map}.)

Property II.) $f$ is such that $U\subset X$ is open whenever $f(U)$ is open.


\item [EXTRA CREDIT] 
 Given a metric space $(X,d)$, let $\tau_d$ be the topology generated by that metric. Consider also a subset $A\subset X$ and $d'=d_A$ the metric $d$ restricted to this subset.
Show that the following two topologies are the same:
$$
\tau_{d,A}, \mbox{ the subspace topology of }\tau_d \mbox{ on } A $$ and $$\tau_{d'}, \mbox{  the metric topology on } A \mbox{ generated by } d'=d_A
$$






\end{enumerate}

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