\chapter{Topological Spaces: Fundamentals}

\begin{thm}    
\label{open-interior}
Let $\{ U_i\}_{i=1}^n $ be a \emph{finite} collection of
open sets in a topological space $(X,{\mathcal T})$. Then $\bigcap_{i=1}^n
U_i$ is open.
 
 
\end{thm}

\oneperpage

\begin{exercise}
Why does your proof not prove the false statement that the infinite intersection of open sets is necessarily open?
\end{exercise}

\oneperpage

\begin{thm}
A set $U$ is open in a topological space $(X,{\mathcal T})$ if and only if for every point $x \in U$, there exists an open set $U_x$ such that $x \in U_x \subset U$.
\end{thm}

\oneperpage

\begin{exercise}
Verify that $\mathcal T_\mathrm{std}$ is a topology on $\R^{n}$; in other words, it satisfies the four conditions of the definition of a topology.
\end{exercise}

\oneperpage

\begin{exercise}
Verify that the discrete, indiscrete, finite complement, and countable complement topologies are indeed topologies for any set $X$. % according to Definition \ref{dfn:topology}.
\end{exercise}

\oneperpage

\begin{exercise}
Describe some of the open sets you get if $\R$ is endowed
with the topologies described above (standard, discrete, indiscrete,
finite complement, and countable complement).  Specifically, identify sets that demonstrate the differences among these topologies, that is, find sets that are open in some topologies but not in others.  For each of the topologies, determine if the interval 
$(0,1)$ is an open set in that topology.
\end{exercise}

\oneperpage

\begin{exercise} 
Give an example of a topological space and a collection of open sets in that topological space that show that the \emph{infinite} intersection of open sets need not be open.
\end{exercise}

\oneperpage

\begin{exercise}
Let $X=\R$ and $A=(1,2)$.  Verify that $0$ is a limit point of $A$ in 
the indiscrete topology and the finite complement topology, but not in the standard
topology nor the discrete topology on $\R$.
\end{exercise}

\oneperpage

\begin{thm}
Suppose $p\not\in A$ in a topological space $(X,\mathcal T)$. Then
$p$ is \emph{not} a limit point of $A$
if and only if there exists 
%an open set $U$ with $p\in U$ 
a neighborhood $U$ of $p$ such that $U\cap A =\emptyset$.
\end{thm}

\oneperpage

\begin{exercise}
If $p$ is an isolated point of a set $A$ in a topological space $X$, then there exists an open set $U$ such that $U\cap A=\{p\}$.
\end{exercise}

\oneperpage

\begin{exercise}
Give examples of sets $A$ in various topological spaces $(X,{\mathcal T})$ with
\be
\item a limit point of $A$ that is an element of $A$;
\item a limit point of $A$ that is not an element of $A$;
\item an isolated point of $A$;
\item a point not in $A$ that is not a limit point of $A$.
\ee
\end{exercise}

\oneperpage

\begin{exercise}\ 
\begin{enumerate}
\item Which sets are closed in a set $X$ with the discrete
topology? 
\item Which sets are closed in a set $X$ with the indiscrete topology? 
\item Which sets are closed in a set $X$ with the finite complement topology?
\item Which sets are closed in a set $X$ with the countable complement topology?
\end{enumerate}
\end{exercise}

\oneperpage

\begin{thm}
For any  topological space $(X,{\mathcal T})$ and $A\subset X$,
$\ol{A}$ is closed.  That is, for any set $A$ in a topological space, 
$\overline{\overline{A}} = \overline{A}$.
\end{thm}

\oneperpage

\begin{thm}
Let $(X,{\mathcal T})$ be a topological space. 
Then the set $A$ is  closed if and only if $X - A$ is open. 
\end{thm}

\oneperpage

\begin{thm}
Let $(X,{\mathcal T})$ be a topological space,
and let $U$ be an open set and $A$ be a closed subset of $X$.
Then the set $U - A$ is open and the set $A - U$ is closed.
\end{thm}

\oneperpage

\begin{thm}
Let $(X, \mathcal{T})$ be a topological space. Then:
\bi
\item[$i$)] $\emptyset$ is closed.
\item[$ii$)] $X$ is closed.
\item[$iii$)]The union of finitely many closed sets is closed.
\item[$iv$)]Let $\{ A_\alpha\}_{\alpha\in\lambda}$ be a
collection of closed subsets in $(X,\mathcal{T})$.  Then
$\bigcap_{\alpha\in\lambda} A_\alpha$ is closed.
\ei
\end{thm}

\oneperpage

\begin{exercise}
Give an example to show that the union of infinitely many closed
sets in a topological space may be a set that is not closed.
\end{exercise}

\oneperpage

\begin{exercise}
Give examples of topological spaces and sets
in them that:
\be
\item are closed, but not open;
\item are open, but not closed;
\item are both open and closed;
\item are neither open nor closed.
\ee
\end{exercise}

\oneperpage

\begin{exercise}
State whether each of the following sets are open, closed, both, or
neither. 
\begin{enumerate}
\item In $\Z$ with the finite complement topology: 
$\{0, 1, 2\}$, $\{ \text{prime numbers} \}$, $\{ n \mid |n| \geq
10\}$. 
\item In $\R$ with the standard topology: $(0,1), (0,1], [0,1], \{0,1\}, 
   \{ 1/n \ | \ n\in \N\}$. 
\item In $\R^2$ with the standard topology: $\{(x,y) \ | \ x^2+y^2 = 1\}$, $\{(x,y) \ | \
x^2+y^2 > 1\}$, $\{(x,y) \ | \ x^2+y^2 \ge 1\}$.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{thm}
\label{smallest-closed-set}
For any set $A$ in a 
topological space $X$, the closure of $A$
equals the intersection of all closed sets containing $A$, that is, 
\[
\overline A=\bigcap_{B\supset A\mathrm{, }B\in\mathcal{C}} B
\]
where $\mathcal{C}$ is the collection of all closed sets in $X$.
\end{thm}

\oneperpage

\begin{exercise}
Pick several different subsets of $\R$, and find their closures in:
\be
\item the discrete topology;
\item the indiscrete topology;
\item the finite complement topology;
\item the standard topology.
\ee
\end{exercise}

\oneperpage

\begin{thm}    Let $A$ and $B$ be subsets of a topological space $X$.  Then
\begin{enumerate}
\item $A\subset B$ implies $\overline{A} \subset \overline{B}$.
\item $\overline{A\cup B} = \overline{A} \cup \overline{B}$.
\end{enumerate}
\end{thm}

\oneperpage

\begin{exercise}
\label{closure-union}
Let $\{A_\alpha\}_{\alpha\in\lambda}$ be a collection of subsets
of a topological space $X$.  
Then is the following statement true?  
$$\overline{\bigcup_{\alpha\in\lambda} A_\alpha} =
\bigcup_{\alpha\in\lambda} \overline A_\alpha.$$
  
\end{exercise}

\oneperpage

\begin{exercise}
In $\R^2$ with the standard topology, describe the limit points and closure of each of the following two sets:
\be
\item 
$S=\left\{\left(x,\sin\left({\frac{1}{x}}\right)\right) \left|\right.  x\in(0,1)\right\}$.
The closure of the set $S$ is called the \emph{topologist's sine curve}\index{topologist's sine curve}.

\item
$C=\{(x,0) \left|\right. x\in[0,1]\}\cup
\bigcup_{n=1}^{\infty}\left\{\left(\frac{1}{n},y\right)\left|\right. y\in[0,1]\right\}$.
The closure of the set $C$ is called the \emph{topologist's comb}\index{topologist's comb}.
\ee
\end{exercise}

\oneperpage

\begin{exercise}
\label{cantor-set-defined}
In the standard topology on $\R$, there exists a non-empty subset $C$ of the closed unit interval $[0,1]$ that is closed, contains no non-empty open interval, and where no point of $C$ is an isolated point.
  
\end{exercise}

\oneperpage

\begin{thm}
 Let $A$ be a subset of a topological space $X$. 
 Then $p$ is an interior point of $A$ if and only if there exists an open set $U$ with $p \in U \subset A$.
\end{thm}

\oneperpage

\begin{exercise}
Show that a set $U$ is open in a topological space $X$ if and only if every point of $U$ is an interior point of $U$.
\end{exercise}

\oneperpage

\begin{thm}
Let $A$ be a subset of a topological space $X$.  
Then $Int(A)$, $Bd(A)$ and $Int(X-A)$ are disjoint sets whose union is $X$.
\end{thm}

\oneperpage

\begin{exercise}
Pick several different subsets of $\R$, and for each one, find its interior and boundary using:
\be
\item the discrete topology;
\item the indiscrete topology;
\item the finite complement topology;
\item the standard topology.
\ee
\end{exercise}

\oneperpage

\begin{thm}
\label{closuresequencethm}
Let $A$ be a subset of the topological space $X$ and let $p$ be a point in $X$.
If $\{ x_i \}_{i\in\N} \subset A$ and $x_i \rightarrow p$, 
then $p$ is in the closure of $A$. 
\end{thm}

\oneperpage

\begin{thm}
In standard topology on $\R^{n}$, if $p$ is a limit point of a set $A$, then there is a sequence of points in $A$ that converge to $p$.
\end{thm}

\oneperpage

\begin{exercise}
Find an example of a topological space and a convergent
sequence in that space, where the limit of the sequence 
is not unique.
\end{exercise}

\oneperpage

\begin{exercise}
\begin{enumerate}
\item Consider sequences in $\R$ with the finite complement topology. Which
sequences converge? To what value(s) do they converge?
\item Consider sequences in $\R$ with the countable complement topology. Which
sequences converge? To what value(s) do they converge?
\end{enumerate}
\end{exercise}

\oneperpage

