\chapter{Connectedness: When Things Don't Fall into Pieces}

\begin{thm}    
The following are equivalent:
\begin{enumerate}
\item $X$ is connected. 
\item there is no  continuous function $f: X\to \R_{std}$ such that $f(X)= \{ 0,1\}$.
\item $X$ is not the union of two disjoint non-empty separated sets.
\item $X$ is not the union  of two disjoint non-empty closed sets.
\item the only subsets of $X$ that are both closed and open in $X$ are the empty set and $X$ itself.
\item for every pair of points $p$ and $q$ and every open cover $\{U_{\alpha}\}_{\alpha \in \lambda}$ of $X$ there exist a finite number of the $U_\alpha$'s, $\{U_{\alpha_1}, U_{\alpha_2}, U_{\alpha_3}, ..., U_{\alpha_n}\}$ such that $p \in U_{\alpha_1}$, $q \in U_{\alpha_n}$ and for each $i < n$, $U_{\alpha_i} \cap U_{\alpha_{i+1}} \neq \emptyset$.
\end{enumerate}
\end{thm}

\oneperpage

\begin{exercise}
Which of the following spaces are connected?
\begin{enumerate}
\item $\R$ with the discrete topology?  
\item $\R$ with the indiscrete topology?  
\item $\R$ with the finite complement topology? 
\item $\R_{LL}$?  
\item $\Q$ as a subspace of $\R_{std}$? 
\item $\R - \Q$ as a subspace of $\R_{std}$?
\end{enumerate}
\end{exercise}

\oneperpage

\begin{thm}    
The space $\R_{std}$ is connected.
\end{thm}

\oneperpage

\begin{thm}    
Let $A,B$ be separated subsets of a space $X$.
If $C$ is a connected subset of $A\cup B$, then either $C\subset A$ or 
$C\subset B$.
\end{thm}

\oneperpage

\begin{thm}    
 Let $\{ C_\alpha\}_{\alpha\in\lambda}$ be a
 collection of connected subsets of $X$ and $E$ be another connected subset
 of $X$ such that for each $\alpha$ in $\lambda$, $E\cap C_\alpha \ne 
 \emptyset$. Then $E\cup \bigl(\bigcup_{\alpha\in\lambda} C_\alpha\bigr)$
 is connected.
 \end{thm}

\oneperpage

\begin{thm}     
Let $C$ be a connected subset of the topological space $X$.  If $D$
is a subset of $X$ such that $C\subset D \subset \overline{C}$, then
$D$ is connected.
\end{thm}

\oneperpage

\begin{exercise}  
Show that the closure of the topologist's sine curve in $\R^2_{std}$ is connected.
\end{exercise}

\oneperpage

\begin{thm}     
Let $X$ be a connected space, $C$ a connected
subset of $X$, and $X-C = A\mid B$.  Then $A\cup C$ and $B\cup C$ are
each connected.
\end{thm}

\oneperpage

\begin{thm}     
For topological spaces $X$ and $Y$, $X\times Y$ is connected
if and only if each of $X$ and $Y$ is connected. 
\end{thm}

\oneperpage

\begin{thm}     
For spaces $\{ X_\alpha\}_{\alpha\in\lambda}$,
$\prod_{\alpha\in\lambda}X_\alpha$ is connected if and only if for
each $\alpha$ in $\lambda$, $X_\alpha$ is connected.
\end{thm}

\oneperpage

\begin{exercise}
Show that the box product of countably infinitely many copies of $\R_{std}$ is not connected. 
\end{exercise}

\oneperpage

\begin{thm}     
Let $f: X \to Y $ be a continuous, surjective function.  If $X$ is connected, 
then $Y$ is connected.
\end{thm}

\oneperpage

\begin{thm}(Intermediate Value Theorem)
\label{thm:ivt}
Let $f:\R_{std} \rightarrow \R_{std}$ be a continuous map.  If $a,b\in \R$ and
$r$ is a point of $\R$ such that $f(a)< r < f(b)$ then there exists a
point $c$ in $(a,b)$ such that $f(c)=r$.
\end{thm}

\oneperpage

\begin{thm}     
\label{thm:countable-regular}
Let $X$ be a countable, regular, $T_1$ space with more than one point.
Then $X$ is not connected.
 \end{thm}

\oneperpage

\begin{exercise}
Show that Bing's Sticky Foot Topology is a countable, connected, Hausdorff space.
\end{exercise}

\oneperpage

\begin{thm}
If $X$ is a normal, $T_1$ space with more than one point and $|X| < |\R|$, then $X$ is not connected.
\end{thm}

\oneperpage

\begin{exercise}     
Let $A$ be a countable subset of $\R^n$ for $n\ge 2$.
Show that $\R^n -A$ is connected. In fact, if the cardinality of $A$ is any cardinality less than the cardinality of $\R$, then $\R^n -A$ will still be connected. Actually, for any two points $p$ and $q$ in $\R^n -A$, $p$ can be connected to $q$ by two intersecting straight line segments in $\R^n -A$.
\end{exercise}

\oneperpage

\begin{thm} 
Each component of $X$ is connected, closed, and not contained in any strictly larger connected subset of $X$.
\end{thm}

\oneperpage

\begin{thm}
The set of components of a space $X$ is a partition of $X$.
\end{thm}

\oneperpage

\begin{lem}
Let $X$ be a topological space and let $\{ H_\alpha\}_{\alpha\in\lambda}$ be the set of subsets of $X$ that are both open and closed. Then the following are equivalent: 
\begin{enumerate}
\item For every two components $A$ and $B$ of $X$, there exists a separation of $X$ into two disjoint closed sets such that $A$ is in one and $B$ is in the other. 
\item For every component $A$ of $X$, $\bigcap \{ H_\alpha \mid A \subset H_\alpha \}= A$. 
\end{enumerate}
\end{lem}

\oneperpage

\begin{lem}
Let $X$ be a compact space and let $U$ be an open set in $X$. Let $\{ H_\alpha\}_{\alpha\in\lambda}$ be closed subsets of $X$ such that 
$\bigcap_{\alpha\in\lambda}H_\alpha \subset U$.  Then there exist a finite number of the $H_\alpha$'s whose intersection lies in $U$.
\end{lem}

\oneperpage

\begin{lem}    
Let $A$ and $B$ be components of a compact,
Hausdorff space $X$. 
Then $X = H\mid K$ where $A\subset H$ and $B\subset K$.
\end{lem}

\oneperpage

\begin{thm}
Let $X$ be a compact, Hausdorff space. Let $X^{*}$ be the partition of $X$ into its components. Then the identification space $X^{*}$ is a compact, Hausdorff space. 
\end{thm}

\oneperpage

\begin{thm}    
Let $A$ and $B$ be closed subsets of a compact,
Hausdorff space $X$ such that no component intersects both $A$ and $B$.
Then $X = H\mid K$ where $A\subset H$ and $B\subset K$.
\label{component}
\end{thm}

\oneperpage

\begin{thm}    
 Let $U$ be a proper, open subset of a 
 continuum $X$.  Then each component of $\overline U$ contains a
 point of $\partial U$, the boundary of $U$.  (Recall: $\partial U = \overline U - U$.)
 \end{thm}

\oneperpage

\begin{thm}
Let $U$ be a proper, open subset of a continuum $X$.  Then each component of $U$ has a limit point on $\partial U$.
 \end{thm}

\oneperpage

\begin{thm}    
 No continuum $X$ is the union of a countable
 number $(>1)$ of disjoint, non-empty closed subsets.
 \end{thm}

\oneperpage

\begin{exercise} 
Show that in Figure \ref{connected-disjoint-sets}, $X$ is connected and is the union of a countable number of disjoint closed sets.
 \end{exercise}

\oneperpage

\begin{thm}    
 Let $\{ C_i\}_{i\in\omega}$ be a collection of
 continua such that for each $i$, $C_{i+1} \subset C_i$.  Then 
 $\bigcap_{i\in\omega} C_i$ is a continuum.
 \end{thm}

\oneperpage

\begin{thm}    
  Let $\{ C_\alpha\}_{\alpha\in\lambda}$ be a 
 collection of continua indexed by a well-ordered set $\lambda$ such that if
 $\alpha <\beta$, then $C_\beta \subset C_\alpha$.  Then $\bigcap_{\alpha
 \in\lambda} C_\alpha$ is a continuum.
 \end{thm}

\oneperpage

\begin{lem}     Let $X$ be a continuum, $p$ be a point of $X$,
 and ${X-\{ p\} = H\mid K}$.  Then $H \cup \{ p\}$ is a continuum and if
 $q\ne p$ is a non-separating point of $H\cup \{ p\}$, then $q$ is a
 non-separating point of $X$.
 \end{lem}

\oneperpage

\begin{thm}
Let $X$ be a separable continuum with more than one point. Then $X$ has at least two non-separating points.
\end{thm}

\oneperpage

\begin{thm}
 Let $X$ be a continuum with more than one point.  Then $X$ has at 
 least two non-separating points.
 \end{thm}

\oneperpage

\begin{thm}
The bucket handle continuum is indecomposable.
\end{thm}

\oneperpage

\begin{thm}    
A path connected space is connected.
\end{thm}

\oneperpage

\begin{exercise}
The flea and comb space is connected but not path connected.
\end{exercise}

\oneperpage

\begin{exercise}
The closure of the topologist's sine curve is connected but not path connected.
\end{exercise}

\oneperpage

\begin{thm}
The product of path connected spaces is path connected.
\end{thm}

\oneperpage

\begin{exercise}
\begin{enumerate}
\item What are the path components of the closure of the topologist's sine curve?
\item What are the path components of the closure of the topologist's comb?
\end{enumerate}
\end{exercise}

\oneperpage

\begin{exercise}
Must every non-empty open connected subset $U$ of $\R^n$ be path connected?
\end{exercise}

\oneperpage

\begin{thm}
Let $p$ and $q$ be two points in a Hausdorff space $X$ such that there exists a continuous function $f:[0,1] \to X$ with $f(0)= p$ and $f(1)=q$. Then there exists an embedding $h:[0,1] \to X$ with $h(0)= p$ and $h(1)=q$.
\end{thm}

\oneperpage

\begin{thm}    
The following are equivalent:
\begin{enumerate}
\item $X$ is locally connected.
\item $X$ has a basis of connected open sets.
\item For each $x \in X$ and open set $U$ with $x \in U$,
the component of $x$ in $U$ is open.
\item For each $x \in X$ and open set $U$ with $x \in U$,
there is a connected set $C$ such that $x \in\hbox{ Int } C \subset
C\subset U$.
%F. I took out /subseteq here.
\item For each $x \in X$ and open set $U$ with $x \in U$,
there is an open set $V$ containing $x$ and $V\subset$ {(the
component of} $x$ in $U)$.
\end{enumerate}
\end{thm}

\oneperpage

\begin{exercise}
\begin{enumerate}
\item Show that the closure of the topologist's comb is not locally connected. 
\item Construct a space that is connected but not locally connected at any point.
\item Find an example of a space that is locally connected but not connected. 
\end{enumerate}
\end{exercise}

\oneperpage

\begin{thm}
The product of two locally connected spaces is locally connected. 
\end{thm}

\oneperpage

\begin{exercise}
\begin{enumerate}
\item Find an example of an infinite number of locally connected spaces where the infinite product space is not connected.
\item Prove that an arbitrary box product of locally connected spaces is locally connected.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{thm}    
Let $X$ be a locally connected space and let $f: X\to Y$ be a continuous, surjective, closed or open map. Then $Y$ is locally connected.
\end{thm}

\oneperpage

\begin{exercise}
Construct an example of a locally connected space $X$ and a continuous, surjective function $f:X \to Y$ such that $Y$ is not locally connected. 
\end{exercise}

\oneperpage

\begin{thm} 
A locally path connected space is locally connected.
\end{thm}

\oneperpage

\begin{thm}    
The following are equivalent:
\begin{enumerate}
\item $X$ is locally arcwise connected.
\item For each $x \in X$ and open set $U$ with $x \in U$,
there is an arcwise connected open set $V$ such that $x \in V \subset U$.
\item $X$ has a basis of connected, arcwise connected open sets.
\end{enumerate}
\end{thm}

\oneperpage

\begin{thm}
A Hausdorff space $X$ is a Peano Continuum if and only if it is the image of $[0,1]$ under a continuous map. 
\end{thm}

\oneperpage

\begin{thm}
Let $f:[0,1] \to X$ be a continuous surjective map where $X$ is Hausdorff. Then $X$ is locally arcwise connected. Equivalently, every Peano Continuum is locally arcwise connected. 
\end{thm}

\oneperpage

\begin{thm}
Let $X$ be a $0$-dimensional, $T_1$ space. Then $X$ is totally disconnected.
\end{thm}

\oneperpage

\begin{exercise}
Create a Hausdorff space that is totally disconnected but is not $0$-dimensional. 
\end{exercise}

\oneperpage

\begin{thm}
The standard Cantor set is precisely those real numbers in $[0,1]$ that can be written using only $0$'s or $2$'s in their ternary (that is, base $3$) expansion.
\end{thm}

\oneperpage

\begin{exercise}
Show that every real number in $[0,2]$ is the sum of two numbers in the standard Cantor set.
\end{exercise}

\oneperpage

\begin{exercise}
Let $C$ be the Cantor set. Create a continuous function $f:C \to [0,1]$ that is surjective.
\end{exercise}

\oneperpage

\begin{exercise}
Let $C$ be the Cantor set. Create an embedding $h:C \to [0,1] \times [0,1]$ such that for every $x \in [0,1]$, $(\{x\} \times [0,1]) \cap h(C) \neq \emptyset$.
\end{exercise}

\oneperpage

\begin{exercise}
Let $C$ be the Cantor set. Create an embedding $h:C \to [-1,1] \times [-1,1] - \{(0,0)\}$ such that every ray from $(0,0)$ straight out to infinity intersects $h(C)$.
\end{exercise}

\oneperpage

\begin{exercise}
Let $C$ be the Cantor set. Create an embedding $h:C \to [0,1] \times [0,1]$ such that for every continuous function $f:[0,1] \to [0,1]$, $G_f \cap h(C) \neq \emptyset$, where $G_f$ is the graph of $f$.
\end{exercise}

\oneperpage

\begin{exercise}
Let $C$ be the Cantor set, let $h:C \to \R^2$ be an embedding, and let $p$ and $q$ be points in $\R^2 - h(C)$. Show that you can find a polygonal path from $p$ to $q$ in $\R^2 - h(C)$.
\end{exercise}

\oneperpage

\begin{thm}
Let $C$ be the standard Cantor set and let $h:C \to \R^2$ be an embedding. Then there exists a homeomorphism $H: \R^2 \to \R^2$ such that for every $x \in C$, $H(h(x))=x$. 
\end{thm}

\oneperpage

\begin{thm}
Let $C$ be the standard Cantor set. There exists an embedding $h:C \to \R^3$ such that no homeomorphism $H: \R^3 \to \R^3$ exists where $H(h(C))=C$. 
\end{thm}

\oneperpage

\begin{thm}
Let $C$ be the standard Cantor set and let $X$ be a $2^{nd}$ countable, compact, Hausdorff space. Then there exists a continuous, surjective function $f:C \to X$. 
\end{thm}

\oneperpage

