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\begin{document}
\title{Homework 2}
\author{Renzo's math 570}
\date{}

\maketitle

Here we go with the second homework set. Do at least $7$ problems, and you must do the problem on the Zariski topology, and exercise $12$!

\section{Another weird topology}

Let $X=\bR$ and define a topology on it:
$$
\tau_{hl}:=\{(a,+\infty)\}_{a\in \bR} \cup\{\phi, X\}
$$

\begin{ex}
Verify that this indeed defines a topology on the real line.
\end{ex}

We call this the \textbf{half line} topology. This is yet another slightly kooky topology that we will use to train ourselves with this abstract nonsense.

\begin{ex}
What are the interior points, the limit points, the interior and the closure for the following sets:
\begin{enumerate}
	\item $V_1=\{0\}$.
	\item $V_2=(0,1)$.
	\item $V_3=(0,+\infty)$.
	\item $V_4=(-\infty,0)$.
\end{enumerate}
\end{ex}

\begin{ex}
For what subsets of $\bR$ is $0$ a limit point?
\end{ex}

\begin{ex}
How does the half line topology compare with the euclidean topology? (is it coarser, finer, neither?)
\end{ex}

Now for some yoga about continuity of functions.

\begin{ex}
Decide if the following statements are true or false:
\begin{enumerate}
	\item If $f:\bR\rightarrow \bR$ is a continuous function when both domain and codomain have the euclidean topology, then $f$ is continuous when the domain has the half line topology and the codomain has the euclidean topology.
	 \item If $f:\bR\rightarrow \bR$ is a continuous function when both domain and codomain have the euclidean topology, then $f$ is continuous when the domain has the euclidean topology and the codomain has the half line topology.
\end{enumerate}
\end{ex}

\begin{ex}
Describe continuous function from $\bR\rightarrow \bR$ when:
\begin{enumerate}
	\item the domain and the codomain have the half line topology.
	\item the domain has the half line topology and the codomain has the euclidean topology.
\end{enumerate}

\end{ex}

\section{Dense Sets}

Let $X$ be a topological space. A set $A$ is called \textbf{dense} if its closure is $X$.
\begin{ex}
Give some example of some dense sets in some familiar topologies. What sets are dense in the discrete and stupid topology?
\end{ex}

Remember the \textbf{finite complement topology} on a set $X$, where the open sets are $\phi$, $X$ and those sets such that their complement is finite.

\begin{ex}
What sets are dense in the finite complement topology? Make sure to discuss both the case where $X$ is a finite or an infinite set.
\end{ex}

\begin{ex}
What sets are dense in the half line topology?
\end{ex}
\begin{ex}[The Zariski Topology]
Consider the following topology on the plane: a set is defined to be closed if it the zero set of a system of  polynomials in two variables, i.e. $C= \{(x,y) \mbox{\ such that \ } f_1(x,y)=f_2(x,y)= \cdots =0\}$.
\begin{enumerate}
	\item Verify that this is indeed a topology on the plane.
	\item Check that all open sets that are not $\phi$ are dense.
	\item Define in the appropriate way the Zariski topology on the line. What topology is it? (i.e. have you seen it before?)
\end{enumerate}
 
\end{ex}

\section{Separating Stuff}

A topological space $X$ is called \textbf{Hausdorff} if, for any pair of points $x,y \in X$, there are two open sets $O_x, O_y$ such that $x \in O_x$, $y \in O_y$ and $O_x\cap O_y =\phi$.

\begin{ex}
Show that if a space $X$ has a \textbf{metrizable} topology, then it is Hausdorff.
\end{ex}

A topological space is called \textbf{T1} if for any pair of points $x,y \in X$, there is an open set $O_x$ such that $x \in O_x$, $y \not\in O_x$.

\begin{ex}
Show that being $T1$ is equivalent to the fact that points are closed sets.
\end{ex}

\begin{ex}
Is $\bR$ with the half line topology a Hausdorff topological space? Is it T1?
\end{ex}
\begin{ex}
Can you think of a topological space that is T1 but not Hausdorff? 
\end{ex}


\end{document}

