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\title{Topology} 
\author{St. Mary's College of Maryland Topology Class}
\begin{document} 

\maketitle

\section{Topological Spaces}
In this chapter we define topological spaces sdfadfg and look at a number of examples.  In addition, we define subspaces and construct corresponding examples.
\begin{defn} A \emph{topological space} is a non-empty set $X$ together with a collection $\TS$ of subsets of $X$ such that:
\begin{enumerate}
\item The empty set is in $\TS$.
\item The set  $X$ itself is in $\TS$.
\item Any \emph{finite} intersection of sets in $\TS$ is also in $\TS$.
\item  \emph{Any } union of sets in $\TS$ is also in $\TS$.
\end{enumerate}
or more formally,
\begin{enumerate}
\item $\emptyset\in \TS$.
\item  $X \in \TS$.
\item  $A_i\in \TS,\ i=1\cdots n \Rightarrow \bigcap \{A_i | i=1\cdots n\} \in \TS$.
\item $A_\alpha\in \TS,\ \alpha\in\Gamma\Rightarrow\bigcup \{ A_\alpha | \alpha\in\Gamma\}\in \TS$.
\end{enumerate}
\end{defn}

 \begin{defn}  Given a topological space $\{X,\TS\}$, $\TS$ is called the \emph{topology}, the elements of $X$ are called \emph{points} and the elements of $\TS$ are called \emph{open sets}.
 \end{defn}

 \begin{defn} A subset $B$ of a topological space $X$ is \emph{closed} $\Leftrightarrow$ $X \setminus B$ (the compliment of $B$) is open.
 \end{defn}

 \begin{lem} If $X$ is a set and $\TS$ a collection of subsets of $X$ such that  $\forall\ A_1, A_2\in \TS, A_1\cap A_2\in \TS$, then axiom 3 is satisfied for $\TS$.
 \end{lem}
\begin{proof}\ \\
\begin{xbox}
Fill in proof
\end{xbox}
\end{proof}

\begin{ex}\ \\
Let $X = \{a,b\}$ and $\TS = \{\emptyset, \{a\}, \{a,b\}\}$, then $\{X,\TS\}$ is a topological space.
\begin{xbox}
Fill in proof
\end{xbox}
\end{ex}

\begin{ex}
 Let $X = \{a,b,c\}$ and $\TS = \{\emptyset, \{a\}, \{b\}, \{a,b,c\}\}$, then $\{X,\TS\}$ is a topological space.\\
\begin{xbox}
Show that this is false, and then make $\TS$ into a topology by adding the fewest possible sets to $\TS$.
\end{xbox}
\end{ex}

\begin{rem} From here on we will use the word \emph{space} to mean a topological space.
\end{rem}

\begin{defn}
(Intuitive version- we'll make it more rigorous later)  Two spaces are \emph{the same} if the points are just renamed.  They are \emph{different} if they are not the same.
\end{defn}

\begin{ex}
Let $X =\{a,b\}$, $\TS_1 = \{\emptyset, \{a\}, \{a,b\}\}$, $\TS_2 = \{\emptyset, \{b\}, \{a,b\}\}$, $\TS_3 = \{\emptyset, \{a,b\}\}$.  \\
\begin{xbox}Describe the relationships between these three topologies.
\end{xbox}
\end{ex}

\begin{ex}
The following are \emph{all} the \emph{different} topologies for $X = \{a,b,c\}$\\
\begin{xbox}Fill this in.
\end{xbox}
\end{ex}

\begin{thm}
Let $X$ be a non-empty set and $\TS=\{\emptyset,X\}$.  Then $\TS$ is a topology for $X$.  
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Fill in proof.
\end{xbox}
\end{proof}

\begin{defn}  We call the topology in the theorem above the \emph{trivial topology} for $X$.
\end{defn}

\begin{thm}
Let $X$ be a non-empty set and $\TS=\{$all subsets of $X\}$.  Then $\TS$ is a topology for $X$.  
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Fill in proof.
\end{xbox}
\end{proof}

\begin{defn}  We call the topology in the theorem above the \emph{discrete topology} for $X$.
\end{defn}

\begin{defn}
Let $\{X,\TS_X\}$ be a space, and $\BS$ a collection of subsets of $X$ such that every element of $\BS$ is in $\TS_X$, and every element of $\TS_X$ can be written as a union of members of $\BS$.  We call $\BS$ a \emph {basis for $\TS_X$}, and $\TS_X$ the \emph{topology induced by $\BS$}.
\end{defn}

\begin{thm}
Let $X$ be a non-empty set and $\BS$ a collection of subsets of $X$ satisfying:
\begin{enumerate}
\item The union of all elements of $\BS$ is X.
\item The intersection of any pair of elements of $\BS$ is the union of elements of $\BS$. 
\end{enumerate}
Let $\mathcal{S}=\{$the set of all possible unions of elements of $\BS\}$.  Then $\mathcal{S}$ is a topology for $X$ with basis $\BS$.
\end{thm}
\begin{proof}\
\begin{xbox}
Fill in proof.
\end{xbox}
\end{proof}

\begin{defn}  Let $\R$ be the set of real numbers, $x \in \R$, and $\epsilon >0$  then \[N(x,\epsilon) = \{y\in\R | x-\epsilon < y < x + \epsilon\}.\]    Note that $N(x,\epsilon)$ can also be written as $\{y\in\R | |x-y|<\epsilon\}$.  We call $N(x,\epsilon)$ an \emph{$\epsilon$-ball about x.} 
\end{defn}

\begin{thm}
Let $X=\R$ and $\BS =\{N(x,\epsilon)|x \in \R,\epsilon>0\}$ then $\BS$ is a basis for a topology on $\R$.
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Fill in proof. Hint:  Use the theorem above.
\end{xbox}
\end{proof}

\begin{defn}\ \\
\begin{xbox}
Extend this idea to to $\R^n$ by defining $N(\bf{x},\epsilon) \subset \R^n=...$
\end{xbox}
\end{defn}

\begin{thm}
Let $X=\R^n$ and $\BS =\{N(\mathbf{x},\epsilon)|\mathbf{x} \in \R ^n, \epsilon >0\}$ then $\BS$ is a basis for a topology on $\R^n$.
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Fill in proof. Hint:  Use the theorem above.
\end{xbox}
\end{proof}


\begin{defn}\ \\
With $\BS$ defined as above we define $\E^n$ to be the topology induced by $\BS$ and refer to $\E^n$ as the \emph{Euclidean topology} for $\R^n$.  We  sometimes just refer to $\E^n$ instead of $\{\R^n, \E^n\}$ for the whole space.
\end{defn}

\begin{thm}
\[\E^n = \{A \subseteq \R^n \ | \forall \mathbf{x} \in A, \exists\ \epsilon > 0 \mathrm{\ such\ that\ }N(\mathbf{x},\epsilon ) \subseteq A\}.\]
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Fill in proof
\end{xbox}
\end{proof}

\begin{defn} Define $\tilde{N}(x,\epsilon) = \{y\in\R | x-\epsilon < y \leq x\}$.
\end{defn}

\begin{thm}
Let $X=\R$ and $\tilde{\BS} =\{\tilde{N}(x,\epsilon)|x \in \R, \epsilon >0\}$ then $\tilde{\BS}$ is a basis for a topology on $\R$.
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Fill in proof.  Use the theorem above..
\end{xbox}
\end{proof}

\begin{defn}  
 We will call $\H$ the \emph{half-open} topology for $\R$.
\end{defn}

\begin{thm}
Let $\F = \{ A \subseteq \R | \R \setminus A$ is a finite set$\} \cup \emptyset$. Then the pair $\{\R, \F\}$ is a topological space.
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Fill in proof
\end{xbox}
\end{proof}

\begin{defn}
We call $\F$ the \emph{Finite-Complement Topology} for $\R$.
\end{defn}

\begin{defn}
Denote $\R$ with the trivial topology as $\{\R, \T\}$, and $\R$ with the discrete topology as $\{\R, \D\}$.
\end{defn}

\begin{ex}\ \\
\begin{xbox}
Determine whether sets consisting of single points are open in any of the five topologies $\E,\H,\F,\T$ and $\D$.
\end{xbox}
\end{ex}

\begin{defn}
Let $X$ be a space, $A\subseteq X$ and $x\in X$.  Then,
$x$ is a \emph{limit point of A} $\Leftrightarrow (\forall O$ open in $X$ such that $x \in O) (O\cap (A\setminus\{x\})\neq\emptyset)$.\\
$Cl(A)$ (the \emph{closure of A}) $=A \cup \{$limit points of $A\}$.\\
\end{defn}

\begin{defn}
Let $X$ be a space, $A\subseteq X$ and $x\in X$.  Then,
$x$ is an \emph{interior point of A} $\Leftrightarrow (\exists O \text{ open in } X \text{ such that }x\in O \subseteq A)$,\\
$Int(A)$ (the \emph{interior of A}) $= \{\text{interior points of } A\}$.
\end{defn}

\begin{ex}
Let $X=\{p,q,r\}, \TS =\{\emptyset, \{r\}, \{p,q\}, \{p,q,r\}\} \text{ and } A=\{p,r\}$.  \\
\begin{xbox}
Determine if any of the points of $X$ are limit points and/or interior points of $A$.
\end{xbox}
\end{ex}

\begin{ex}
Let $X=\E^2$ (really, $X=\R^2, \TS_X=\E^2$) and $A=\{(x,y)|0\le x<2, 0\le y<2\}$.  \\
\begin{xbox}
Determine if any of the points $(0,0), (1,1), (1,2), (3,3) $ of $X$ are limit points and/or interior points of $A$.
\end{xbox}
\end{ex}

\begin{thm}
In any space $X$, a set $A\subseteq X$ is open $\Leftrightarrow$ All points of $A$ are interior.
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Fill in proof.
\end{xbox}
\end{proof}

\begin{thm}
In any space $X$, a set $A\subseteq X$ is closed $\Leftrightarrow A$ contains all of its limit points.
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Fill in proof.
\end{xbox}
\end{proof}

\begin{defn}
Let $X$ be a space with topology $\TS_X$, and $Y\subseteq X$ where $Y\neq\emptyset$.  Let $\TS_Y = \{A\subseteq Y|\ \exists\ O \text{ open in } X (ie. O \in \TS_X) \text{ with } A=O\cap Y\}$.
\end{defn}

\begin{thm} 
The collection $\TS_Y$ is a topology for $Y$.
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Fill in proof
\end{xbox}
\end{proof}

\begin{defn}
With this topology, we call $Y$ a \emph{subspace} of $X$ and we call $\TS_Y$ the \emph{subspace topology} for $Y$.
\end{defn}

\begin{ex}
Let $X=\{a,b,c,d\}, \TS_X= \{\emptyset, \{a\}, \{a,b\}, \{a,c\}, \{a,b,c\},\{a,b,c,d\}\}$, and $Y=\{a,b,d\}$.  Give $Y$ the subspace topology.\\
\begin{xbox}
List the open sets in $Y$\\
Give examples of non-empty subsets of $Y$ that are:
\begin{enumerate}
\item open in $Y$ and open in $X$. 
\item open in $Y$ but not open in  $X$.  
\item open in $X$ but not open in $Y$.
\end{enumerate}
\end{xbox}
\end{ex}

\begin{ex}
Let $X=\E^2$ and $Y=\{(x,y)|0\le x<1, 0\le y<1\}$.  Give $Y$ the subspace topology.\\  
\begin{xbox}
Give examples of non-empty subsets of $Y$ that are:
\begin{enumerate}
\item open in $Y$ and open in $X$. 
\item open in  $Y$ but not open in  $X$.  
\item open in $X$ but not open in $Y$.
\end{enumerate}
\end{xbox}
\end{ex}

\begin{thm}\ \\
\begin{xbox}
Make up a simple theorem about open subsets in subspaces.
\end{xbox}
\end{thm}
\begin{proof}\ \\
\begin{xbox}
Fill in proof
\end{xbox}
\end{proof}

\begin{defn}
Let $X= \E^n$ and $Y=\{(x_1, x_2, \cdots, x_n)|x_1^2+x_2^2+\cdots+x_n^2=1\}$.  Give $Y$ the subspace topology.  We will refer to $Y$ as $\S^{n-1}$.\\
If we change the '=' to '$\le$', the resulting space is called $\B^n$.
\end{defn}

\begin{ex}\ \\
\begin{xbox}
Draw pictures of $\S^{n-1}$ and $\B^n$ for $n=1,2,3$.  In your pictures draw some representative open sets.  Can you guess why we use $\S^{n-1}$ instead of $\S^n$?
\end{xbox}
\end{ex}


 \end{document}