generalNotes.tex

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\textbf{Table of Contents}
\begin{itemize}
  \item[] \hyperref[sec:Definitions]{Definitions and Theorems}
  \begin{itemize}
    \item[] \hyperref[sec:EarlyChapters]{Early Chapters}
    \item[] \hyperref[sec:continuity]{Continuity}
    \item[] \hyperref[dfn:cartesianProducts]{Cartesian Products}
    \item[] \hyperref[dfn:metric]{Metrics}
    \item[] \hyperref[dfn:quotientMapTopology]{Quotient Maps, Spaces, and Topology}
    \item[] \hyperref[sec:connectedness]{Connected Spaces}
    \item[] \hyperref[sec:connectedRealLine]{Connected Subspaces of the Real Line}
  \end{itemize}
  \item[]  Exercises
  \begin{itemize}
    \item[] \hyperref[sec:chapter1.7]{Chapter 1 Section 7}
    \item[] \hyperref[sec:chapter2.13]{Chapter 2 Section 13}
    \item[] \hyperref[sec:chapter2.16]{Chapter 2 Section 16}
    \item[] \hyperref[sec:chapter2.17]{Chapter 2 Section 17}
    \item[] \hyperref[sec:chapter2.18]{Chapter 2 Section 18}
    \item[] \hyperref[sec:chapter2.19]{Chapter 2 Section 19}
    \item[] \hyperref[sec:chapter2.20]{Chapter 2 Section 20}
    \item[] \hyperref[sec:chapter2.21]{Chapter 2 Section 21}
    \item[] \hyperref[sec:chapter2.22]{Chapter 2 Section 22}
    \item[] \hyperref[sec:chapter3.23]{Chapter 3 section 23}
    \item[] \hyperref[sec:chapter3.24]{Chapter 3 section 24}
  \end{itemize}
\end{itemize}
\noindent \textbf{Definitions:} \label{sec:Definitions}
\begin{enumerate}
  \item \label{sec:EarlyChapters} Basics, before continuity. \begin{itemize}
    \item \label{dfn:functionRestriction} $f|A$: The function $f$ restricted to the domain $A$.
    \item \label{dfn:simpleOrder} A relation $C$ is an order, simple order, or linear order if it is comparable: for every distinct $x,y$, either $xCy$ or $yCx$; nonreflexive: for no $x$ is $xCx$ true; and transitive, if $xCy$ and $yCz$ then $xCz$.
    \item \label{dfn:leastUpperBoundProperty} Least Upper Bound Property: Any set that has an upper bound has a least upper bound.
    \item \label{thm:AxiomChoice} The axiom of choice: given a collection $\mathcal{A}$ of disjoint nonempty sets, there exists a set $C$ consisting of exactly one element from each element of $\mathcal{A}$; that is, a set $C$ such that $C$ is contained within the union of elements of $\mathcal{A}$, and for each $A\in\mathcal{A}$, the set $C\cap A$ contains a single element.
    \item \label{dfn:finiteComplementTopology} The finite complement topology $\T_f$: Let $X$ be a set, then $\T_f$ is the collection of all subsets $U$ of $X$ such that $X-U$ is either finite or all of $X$
    \item \label{dfn:finer} Finer: $\T_1$ is finer than $\T_2$ implies that $\T_2\subseteq \T_1$, in other words, $\T_1$ has all the open sets of $\T_2$ and more.
    %previous is at most section 12, now at least section 13
    \item \label{dfn:basis} A basis for a topology on $X$ is a collection $\mathcal{B}$ of subsets of $X$ (called basis elements) such that for each $x\in X$ there is at least one basis element containing $x$, and if $x$ belongs to the intersection of two basis elements $B_1$ and $B_2$, then there is a basis element $B_3$ containing $x$ such that $B_3 \subset B_1 \cap B_2$. Then the topology generated by the basis is: a subset $U$ of $X$ is open if for each $x\in U$ there is a basis element $B$ such that $x\in B$ and $B\subset U$.
    \begin{itemize}
      \item \label{thm:basisUnion} Equivalently, given $X$ and a basis $\mathcal{B}$, the topology generated by the basis is the collection of all unions of elements of $\mathcal{B}$.
      \item \label{thm:basisFiner} Also, $\T'$ is finer that $\T$ if and only if for each $x\in X$ and each basis element $B\in\mathcal{B}$ containing $x$, there is a basis element $B'\in \mathcal{B}'$ such that $x\in B'\subset B$.
    \end{itemize}
    \item \label{dfn:subbasis} Subbasis: a subbasis $\mathcal{S}$ for a topology on $X$ is a collection of subsets of $X$ whose union equals $X$. The topology generated by the subbasis is defined to be the collection of all unions of finite intersections of elements of $\mathcal{S}$.
    \item \label{dfn:lowerLimitTopology} The lower limit topology on $\R$, denoted $\R_\ell$ is defined as the topology with the basis that is the collection of all half open intervals of the form $[a,b) = \{ x\; |\; a\leq x < b \}$
    \item \label{dfn:KTopology} The $K$-Topology on $\R$ is defined as follows. First, let $K$ denote the set of all numbers $1/n$, for $n\in\mathbb{Z}_+$. The $K$-topology is generated by the basis that is defined by the collection of all open intervals $(a,b)$, along with all sets of the form $(a,b)-K$, with $a,b\in\R$.
    % previous is section 13, after is at least section 14
    \item \label{dfn:orderTopology} Order Topology: If $S$ is a totally ordered set, then let $\chi$ be the basis made of open rays $S$.
    \item \label{dfn:dictionaryOrder} Dictionary order for $\R\times\R$ is defined by the basis that is the collection of all open intervals $(a\times b, c\times d)$ where $a<c$, or both $a=c$ and $b<d$.
    \item \label{dfn:orderedSquare} The ordered square, $I_o^2$ is defined as $[0,1]\times [0,1]$ in the dictionary order topology.
    \item \label{dfn:projection} Projection: for example, $\pi_1: X\times Y \rightarrow X$ is defined by $\pi_1(x,y)=x$, and $\pi_2(x,y)=y$.
    \item \label{dfn:subspace} Subspace Topology: $Y$ is the \emph{subspace} of $X$ if $Y$ is a subset of $X$, and $X$ is a topological space with topology $\T$, and the \emph{subspace topology} on $Y$ is $\T_Y=\{Y\cap U | U\in \T$.
  \end{itemize}
  \item \label{sec:closedSetsLimitPointsHausdorff} Closed Sets, Limit Points, Hausdorff Spaces - Chapter 2, Section 17 \begin{itemize}
    \item \label{dfn:intersects} Intersects: $A$ intersects $B$ if $A \cap B \neq \emptyset$
    \item \label{dfn:neighborhood} Neighborhood: If $U$ is an open set containing $x$, then $U$ is a neighborhood of $x$.
    \item \label{dfn:interior} Interior of $A$ ($\text{Int }A$): The union of all open sets contained in $A$. The interior of an open set is itself.
    \item \label{dfn:closure} Closure of $A$ ($\bar{A}$): The intersection of all the closed sets containing $A$. Equivalently, $A$ together with all of its limit points. The closure of a closed set is itself.
    \item \label{dfn:boundary} Boundary: $\text{Bd } A = \bar{A}\cap(\overline{X-A})$. Also, $\bar{A} = \text{Int } A \cup \text{Bd } A$. The interior and the boundary of $A$ are disjoint, and the boundary is empty if and only if $A$ is open and closed. $U$ is open if and only if the boundary is $\bar{U}-U$.
    \item \label{dfn:limitPoint} Limit Point: $x$ is a limit point of $A$ if and only if every neighborhood $U$ of $x$ contains some point $y\in A$ distinct from $x$.
    \item \label{thm:BelongsClosure17.5} Theorem 17.5:  $x\in \bar{A}$ if and only if every open set $U$ or basis element $B$ containing $x$ intersects $A$.
    \item \label{dfn:converge} Convergence: A sequence $x_1, x_2, ...$ converges to $x$ if for every neighborhood $U$ of $x$ there exists some $m_U\in\mathbb{N}$ such that $\forall n>m_U\; x_n\in U$.
    \item \label{dfn:Hausdorff} Hausdorff Space: If for each pair of distinct points $x_1,x_2$ of a topologocal space $X$ there exist \emph{disjoint} neighborhoods $U_1$ and $U_2$ of $x_1$ and $x_2$ respectively, then the space is Hausdorff.
    \begin{itemize}
      \item \label{thm:FinitePointHausdorff} Every finite point set in a Hausdorff space is closed. (This is weaker than Hausdorff, on it's own, it is the \textbf{$T_1$ axiom.}
      \item \label{thm:LimitPointT1} $X$ satisfies the $T_1$ axiom, $A$ is a subset. Then, the point $x$ is a limit point of $A$ if and only if every neighborhood of $x$ contains infinitely many points of $A$.
      \item \label{thm:inheritingHausdorff} Every simply ordered set is Hausdorff in the order topology, the product of two Hausdorff spaces is Hausdorff, and a subspace of a Hausdorff space is Hausdorff.
    \end{itemize}
    %End of section 17
  \end{itemize}
  \item \label{sec:continuity} Continuity. Chapter 2, section 18
  \begin{itemize}
    \item \label{dfn:continuous} Continuous: A function $f: X\rightarrow Y$ is said to be continuous if for each open subset $V$ of $Y$, the set $f^{-1}(V)$ is an open subset of $X$. It is sufficient to show that the inverse image of every basis element, or even of every subbasis element is open.
    Equivalently:
    \begin{enumerate} %Theorem 18.1
      \item \label{dfn:continuous2} For every subset $A$ of $X$, $f(\bar{A}) \subset \overline{f(A)}$
      \item \label{dfn:continuous3} For every closed set $B$ of $Y$, the set $f^{-1}(B)$ is closed in $X$.
      \item \label{dfn:continuous4} For each $x\in X$ and each neighborhood $V$ of $f(x)$, there is a neighborhood $U$ of $x$ such that $f(U) \subset V$
    \end{enumerate}
    Note: if the third condition holds for one point, then $f$ is continuous at that point.
    \item \label{dfn:homeomorphism} Homeomorphism: A bijective function $f$ is a homeomorphism is both $f$ and $f^{-1}$ are continuous.
    Equivalently, $f(U)$ is open if and only if $U$ is open.
    \item \label{dfn:embedding} Embedding: Let $f: X\rightarrow Y$ be a continuous injective map, and set $Z=f(X)$ considered as a subspace of $Y$; then the function $f': X\rightarrow Z$ obtained by restricting the range of $f$ is bijective. If $f'$ is a homeomorphism, then we say $f:X\rightarrow Y$ is an embedding of $X$ in $Y$.
    \item \label{thm:PastingLemma} Pasting Lemma: let $X=A\cup B$ where $A$ and $B$ are closed int $X$. Let $f: A\rightarrow Y$ and $g: B\rightarrow Y$ be continuous. If $f(x)=g(x)$ fo every $x\in A\cap B$ then $f$ and $g$ combine to give a continuous function $h: X\rightarrow Y$ defined by setting $h(x) = f(x)$ if $x\in A$ and $h(x) = g(x)$ if $x\in B$.
    \item \label{thm:LocalFormulationContinuity} The local formulation of continuity: If $X$ can be written as the union of upen sets $U_\alpha$ such that $f|U_\alpha$ is continuous for each $\alpha$, then the map $f: X\rightarrow Y$ is continuous.
    \item \label{thm:ChangingRangeDomainContinuity} Restricting the domain, or restricting or expanding the range of a continuous function will yield a continuous function.
    \item \label{thm:MapsProducts18.4} Theorem 18.4 - Maps into products. Let $f : A\rightarrow X\times Y$ be given by $f(a)=(f_1(a),f_2(a))$. Then $f$ is continuous if and only if $f_1$ and $f_2$ (the coordinate functions of $f$) are continuous.
  \end{itemize}
  \item \label{dfn:cartesianProducts} Cartesian Products, chapter 2 section 19
  \begin{itemize}
    \item $\mathcal{A}$ is a nonempty collection of sets. Indexing function for $\mathcal{A}$ is a surjective function $f$ from the index set, $J$, to $\mathcal{A}$. Given $\alpha \in J$, the set $f(\alpha)$ will be denoted $A_\alpha$, the indexed family will be $\{A_\alpha\}_{\alpha\in J}$ or $\{A_\alpha\}$.
    \item Arbitrary unions/intersections: $\displaystyle\bigcup_{\alpha\in J} A_\alpha = \{x |\; \exists \alpha \in J \text{ such that } x\in A_\alpha \}$. $\displaystyle\bigcap_{\alpha\in J} A_\alpha = \{x |\; \forall \alpha \in J,\; x\in A_\alpha \}$.
    \item $m$-Tuples: an $m$-tuple is the function $x: (1,\dots,m)\rightarrow X$. $x(i)$ is typically denoted $x_i$ and called the $i$th coordinate of $x$, and $x$ is denoted $(x_1\dots x_m)$.
    \item Cartesian product. $\{A_1\dots A_m\}$ is a family of sets, indexed by the set $\{1\dots m\}$. $X = A_1 \cup\dots \cup A_m$. The cartesian product $\displaystyle\prod_{i=1}^{m}A_i$ is the set of all $m$-tuples $(x_1,\dots ,x_m)$ of elements of $X$ such that $\forall i\; x_i \in A_i$
    \item $J$-tuple: given an index set $J$ a $J$-tuple is a function $x: J\rightarrow X$. For any $\alpha\in J$, we denote $x(\alpha)$ by $x_\alpha$, we call it the $\alpha$-th coordinate of $x$, and denote the function $x$ by the symbol $(x_\alpha)_{\alpha\in J}$. The set of all $J$-tuples of elements of $X$ is denoted $X^J$.
    \item Cartesian product. $\{A_\alpha\}_{\alpha\in J}$ is an indexed family of sets, $X=\bigcup_{\alpha\in J}A_\alpha$. The cartesian product, $\prod_{\alpha\in J} A_\alpha$ is defined as the set of all $J$-tuples of elements of $X$ such that $x_\alpha \in A_\alpha$ for each $\alpha\in J$. Also denoted $\prod A_\alpha$, and its general element as $(x_\alpha)$ if the index set is understood.
    \item Box Topology: let $(X_\alpha)_{\alpha\in J}$ be an indexed family of topological spaces. Then give the product space $\displaystyle\prod_{\alpha\in J}X_\alpha$ the basis that is the collection of all sets of the form $\displaystyle\prod_{\alpha\in J}U_\alpha$, where $U_\alpha$ is open in $X_\alpha$ for all $\alpha\in J$.
    \item Product topology: Let $\pi_\beta : \displaystyle\prod_{\alpha\in J}X_\alpha \rightarrow X_\beta$ be defined $\pi_\beta((x_\alpha)_{\alpha\in J}) = x_\beta$; it maps each element of the product space to its $\beta$th coordinate, it is the projection mapping associated with index $\beta$. \\
    Let $\mathcal{S}_\beta$ denote the collection $\mathcal{S}_\beta = \{\pi_\beta^{-1}(U_\beta) | U_\beta \text{ is open in } X_\beta\}$. Then let $\mathcal{S}$ denote the union $\mathcal{S}=\displaystyle\bigcup_{\beta\in J}\mathcal{S}_\beta$. The topology generated by subbasis $\mathcal{S}$ is called the product topology, and $\prod_{\alpha\in J} X_\alpha$ is a product space.\\
    From this, we get the basis of the product topology: all sets of the form $\prod U_\alpha$, where $U_\alpha$ is an open set of $X_\alpha$, and is equal to $X_\alpha$ for all but finitely many values of $\alpha$.
    \item \label{thm:MapsProducts19.6} Theorem 19.6: Let $f: A \rightarrow \prod_{\alpha\in J}X_\alpha$ be defined $f(a) = (f_\alpha(a))_{\alpha \in J}$, where $f_\alpha : A \rightarrow X_\alpha$ for each $\alpha$. Let $\prod X_\alpha$ have the product topology. Then $f$ is continuous if and only if each $f_\alpha$ is continuous.
  \end{itemize}
  \item \label{dfn:metric} Metrics, chapter 2 sections 20,21
  \begin{itemize}
    \item A metric is a function $d: X \times X \rightarrow \mathbb{R}$ such that: $d(x,y)>0$ for $x\neq y$, $d(x,x)=0$; $d(x,y)=d(y,x)$; and $d(x,y) + d(y,z)\geq d(x,z)$.
    \item The $\epsilon$-ball centered at $x$ is $B_d(x, \epsilon) = \{ y | d(x,y) < \epsilon \}$.
    \item \label{dfn:metricTopology} The metric topology induced by a metric $d$ on a set $X$ is defined by the basis consisting of all $\epsilon$-balls $B_d(x,\epsilon)$ for all $x\in X$ and $\epsilon > 0$. Alternatively, a set $U$ is open in the metric topology induced by $d$ if and only if for each $y\in U$, there is a $\delta > 0$ such that $B_d(y,\delta) \in U$.
    \item A space is metrizable of there exists a metric on it that induces its topology. A metric space is a metrizable space together with the metric that gives it its topology.
    \item Bounded: A subset $A$ of $X$ is bounded if there exists an $M$ such that for all points $x,y$ $d(x,y)<M$. If $A$ is bounded and nonempty, the diameter of $A$ is defined as $\sup( \{d(x,y) | x,y\in X\})$
    \item Standard Bounded Metric $\bar{d}$: Let $X$ be a metric space with metric $d$. Define $\bar{d}: X\times X \rightarrow \mathbb{R}$ by the equation $\bar{d}(x,y) = \min(d(x,y),1)$. $\bar{d}$ induces the same topology as $d$.
    \item Euclidean and Square Metrics on $\mathbb{R}^n$: The euclidean metric $d(x,y) = ||x-y|| = ((x_1 - y_1)^2 + \dots + (x_n - y_n)^2)^{1/2}$. The square metric $p(x,y) = \max(|x_1 - y_1|, \dots , |x_n - y_n|)$.
    \item Uniform Metric $\bar{p}$ on $\mathbb{R}^J$, inducing the uniform topology: $\bar{p}(x,y) = \sup(\{\bar{d}(x_\alpha, y_\alpha) | \alpha \in J\})$. The uniform topology is finer than the product topology and coarser than the box topology, and different if $J$ is infinite.
    \item \label{dfn:countableBasis} Countable Basis at Point $x$: one exists if there is a countable collection $\{U_n\}_{n\in\mathbb{Z}_+}$ of neighborhoods of $x$ such that any neighborhood $U$ of $x$ contains at least one of the sets $U_n$. A space that has a countable basis at each of its points satisfies the \emph{first countability axiom}. All metrizable spaces satisfy this axiom.
    \item \label{dfn:convergesUniformly} Converging Uniformly: Let $(f_n): X\rightarrow Y$ be a sequence of functions where $Y$ is a metric space with metric $d$. The sequence converges uniformly to the function $f:X\rightarrow Y$, if given any $\epsilon>0$ there exists an integer $N$ such that $d(f_n(x), f(x))<\epsilon$ for all $n>N$ and for all $x\in X$.  If $(f_n)$ converges uniformly to $f$ and each $f_n$ is continuous, then $f$ is continuous.
    \item \label{thm:SequenceContinuity21.3} Theorem 21.3 and Lemma 21.2 (The Sequence Lemma) - If there is a sequence of points of $A$ converging to $x$, then $x\in \bar{A}$, the converse holds if $X$ has a countable basis at point $x$, or, more strongly, if $X$ is metrizable. If a function $f$ is continuous, then for every convergent sequence $x_n\rightarrow x$ in $X$, the sequence $f(x_n)$ converges to $f(x)$, again the converse holds if $X$ has a countable basis at $x$. 
  \end{itemize}
  \item \label{dfn:quotientMapTopology} Quotient Topology, chapter 2 section 22
  \begin{itemize}
    \item \label{dfn:saturated} Saturated: A subset $C$ of $X$ is saturated with respect to a surjective map $p: X\rightarrow Y$ if $C$ contains every set $p^{-1}(\{y\})$ that it intersects. 
    \item \label{dfn:quotientMap} Quotient Map: let $p: X\rightarrow Y$ be a surjective map. $p$ is a quotient map when a subset $U$ of $Y$ is open if and only if $p^{-1}(U)$ is open in $X$, which is stronger than continuity. Equivalently, replace 'open' by 'closed'. Alternatively, $p$ is a quotient map if $p$ is continuous and $p$ maps saturated open sets of $X$ to open sets of $Y$, or saturated closed sets of $X$ to closed sets of $Y$.
    \item \label{dfn:OpenClosedMap} An open map $f:X\rightarrow Y$ is one that maps every open set of $X$ to an open set of $Y$, and a closed map sends every closed set of $X$ to a closed set of $Y$. If $f$ is a surjective open or closed map, $f$ is a quotient map, but some quotient maps are neither open nor closed.
    \item \label{dfn:quotientTopology} Quotient Topology: If $X$ is a space and $A$ is a set and if $p: X\rightarrow A$ is a surjective map, then there exists exactly one topology $\T$ on $A$ relative to which $p$ is a quotient map; $\T$ is the quotient topology induced by $p$.
    \item \label{dfn:quotientSpace} Quotient Space: Let $X$ be a topological space and let $X^*$ be a partition of $X$ into disjoint subsets whose union is $X$. Let $p: X\rightarrow X^*$ be the surjective map that carries each element of $X$ to the element of $X^*$ that contains it. In the quotient topology induced by $p$, the space $X^*$ is called the quotient space of $X$. Often called the identification space or decomposition space of $X$. The typical open set of $X^*$ is a collection of equivalence classes whose union is open in $X$.
    \item \label{thm:restrictionQuotientMap} Theorem 22.1 Restricting Quotient Maps: Let $p: X\rightarrow Y$ be a quotient map; let $A$ be a subspace of $X$ that is saturated with respect to $p$; let $q: A\rightarrow p(A)$ be the map obtained by restricting $p$. If $A$ is either open or closed in $X$, then $q$ is a quotient map. If $p$ is either an open or closed map, then $q$ is a quotient map.
    \item \label{thm:quotientMapGeneralProperties} The restriction of a quotient map need not be a quotient map. The composition of two quotient maps is a quotient map. The cartesian product of two quotient maps need not be a quotient map, unless both maps are open, or the spaces are locally compact (defined later). If $X$ is Hausdorff, the quotient space $X^*$ need not be Hausdorff.
    \item \label{thm:continuityQuotientMap} Theorem 22.2 Continuity of quotient maps: Let $p: X\rightarrow Y$ be a quotient map. Let $Z$ be a space and let $g\rightarrow X\rightarrow Z$ be a map that is constant on each set $p^{-1}(\{y\})$, for $y\in Y$. Then $g$ induces a map $f:Y\rightarrow Z$ such that $f\circ p = g$. The induced map $f$ is continuous if and only if $g$ is continuous; $f$ is a quotient map if and only if $g$ is a quotient map.
    \item \label{thm:HausdorffHomeomorphismQuotientMap} Let $g: X\rightarrow Z$ be a surjective continuous map. Let $X^*$ be the following collectino of subsets of $X$: $X^*=\{g^{-1}(\{z\}) | z\in Z\}$. Give $X^*$ the quotient topology. If $Z$ is Hausdorff, so is $X^*$. The map $g$ induces a bijective continuous map $f: X^*\rightarrow Z$, which is a homeomorphism if and only if $g$ is a quotient map.
  \end{itemize}
  \item \label{sec:connectedness} Connected Spaces, chapter 3 section 23
  \begin{itemize}
    \item \label{dfn:separation} Separation: Let $X$ is a topological space. A separation of $X$ is a pair $U,V$ of disjoint nonempty open subsets of $X$ whose union is $X$.
    \item \label{dfn:connected} Connected: A space $X$ is connected if there exists no separation of $X$, or equivalently if and only if the subsets of $X$ that are both open and closed in $X$ are the empty set and itself. Any space homeomorphic to a connected space is connected.
    \item \label{dfn:subspaceSeparation} Separation of a subspace - Lemma 23.1: If $Y$ is a subspace of $X$, a separation of $Y$ is a pair of disjoint nonempty sets $A$ and $B$ whose union is $Y$, neither of which contains a limit point of the other.
    \item \label{thm:subspaceOfConnected} Subspace of a Connected Space - Lemma 23.2: If $C$ and $D$ form a separation of $X$ and $Y$ is a connected subspace of $X$, then $Y$ lies entirely in $C$ or $D$.
    \item \label{thm:unionConnected} Union of connected spaces, Theorem 23.3: The union of a collection of connected subspaces of $X$ that have a point in common is connected.
    \item \label{thm:closureConnected} Closure of a connected space: Let $A$ be a connected subspace of $X$. If $A\subset B\subset \bar{A}$, then $B$ is also connected. Equivalently, if $B$ the union of $A$ and some or all of its limit points, then $B$ is connected.
    \item \label{thm:continuousConnected} Theorem 23.5: The image of a connected space under a continuous map is connected.
    \item \label{thm:finiteCartesianConnected} Theorem 23.6: A finite cartesian product of connected spaces is connected. The arbitrary product is connected in the product topology.
  \end{itemize}
  \item \label{sec:connectedRealLine} Connected Subspaces of the Real Line, chapter 3 section 24
  \begin{itemize}
    \item \label{dfn:linearContinuum} Linear Continuum: a \hyperref[dfn:simpleOrder]{simply ordered} set $L$ having more than one element, the \hyperref[dfn:leastUpperBoundProperty]{least upper bound property}, and the property that if $x<y$, then there exists a $z$ such that $x<z<y$.
    \item \label{thm:continuumConnected} Theorem 24.1: A linear continuum, and rays and intervals within it are connected. $\R$ is connected.
    \item \label{thm:IntermediateValue} Intermediate Value Theorem, 24.3: Let $f:X\rightarrow Y$ be a continuous map, where $X$ is a connected space and $Y$ is an ordered set in the order topology. If $a$ and $b$ are two points of $X$ and if $r$ is a point of $Y$ lying between $f(a)$ and $f(b)$, then there exists a point $c$ of $X$ such that $f(c)=r$.
    \item \label{dfn:pathConnected} Path and Path Connected: A path in $X$ from point $x$ to point $y$ of $X$ is a continuous map $f:[a,b]\rightarrow X$ of some closed interval of the real line into $X$, such that $f(a)=x$ and $f(b)=y$. A space $X$ is said to be path connected if every pair of points of $X$ can be joined by a path in $X$.
  \end{itemize}
\end{enumerate}
\textbf{Chapter 1.7} \label{sec:chapter1.7}
\input{./ch1.7/exercises.tex}
%\textbf{Chapter 1.10}
%\input{./ch1.10/exercises.tex}
\textbf{Chapter 2.13} \label{sec:chapter2.13}
\input{./ch2.13/exercises.tex}
\textbf{Chapter 2.16} \label{sec:chapter2.16}
\input{./ch2.16/exercises.tex}
\textbf{Chapter 2.17} \label{sec:chapter2.17}
\input{./ch2.17/exercises.tex}
\textbf{Chapter 2.18} \label{sec:chapter2.18}
\input{./ch2.18/exercises.tex}
\textbf{Chapter 2.19} \label{sec:chapter2.19}
\input{./ch2.19/exercises.tex}
\textbf{Chapter 2.20} \label{sec:chapter2.20}
\input{./ch2.20/exercises.tex}
\textbf{Chapter 2.21} \label{sec:chapter2.21}
\input{./ch2.21/exercises.tex}
\textbf{Chapter 2.22} \label{sec:chapter2.22}
\input{./ch2.22/exercises.tex}
\textbf{Chapter 3.23} \label{sec:chapter3.23}
\input{./ch3.23/exercises.tex}
\textbf{Chapter 3.24} \label{sec:chapter3.24}
\input{./ch3.24/exercises.tex}
\end{document}