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\begin{document}

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\section*{pointed topological space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{the_category_of_pointed_topological_spaces}{The category of pointed topological spaces}\dotfill \pageref*{the_category_of_pointed_topological_spaces} \linebreak
\noindent\hyperlink{ForgettingAndAdjoiningBasepoints}{Forgetting and adjoining basepoints}\dotfill \pageref*{ForgettingAndAdjoiningBasepoints} \linebreak
\noindent\hyperlink{wedge_sum_and_smash_product}{Wedge sum and Smash product}\dotfill \pageref*{wedge_sum_and_smash_product} \linebreak
\noindent\hyperlink{mapping_cocones}{Mapping (co-)cones}\dotfill \pageref*{mapping_cocones} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{pointed topological space} (often \emph{[[pointed space]]}, for short) is a [[topological space]] equipped with a choice of one of its [[points]] ([[elements]]).

Although this concept may seem simple, pointed topological spaces play a central role for instance in [[algebraic topology]] as domains for [[reduced cohomology|reduced]] [[generalized (Eilenberg-Steenrod) cohomology theories]] and as an ingredient for the definition of [[spectra]].

One reason why pointed topological spaces are important is that the [[category]] which they form is an intermediate stage in the [[stabilization]] of [[homotopy theory]] (the [[classical model structure on topological spaces|classical homotopy theory of topological spaces]]) to [[stable homotopy theory]]:

The category of pointed topological spaces has a [[zero object]] (the [[point space]] itself) and the canonical [[tensor product]] on pointed spaces is the \emph{[[smash product]]}, which is non-[[cartesian monoidal category]], in contrast to the plain [[product space|product of topological space]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \emph{pointed topological space} is a [[topological space]] $(X,\tau)$ equipped with a choice of point $x \in X$. A [[homomorphism]] between pointed topological space $(X,x)$ $(Y,y)$ is a [[continuous function]] $f \colon X \to Y$ which preserves the chosen basepoits in that $f(x) = y$.

\hypertarget{the_category_of_pointed_topological_spaces}{}\subsubsection*{{The category of pointed topological spaces}}\label{the_category_of_pointed_topological_spaces}

Stated in the language of [[category theory]], this means that pointed topological spaces are the \emph{[[pointed objects]]} in the [[category]] [[Top]] of topological spaces. This is the [[coslice category]] $Top^{\ast/}$ of topological spaces ``under'' the [[point space]] $\ast$:

an [[object]] in $Top^{\ast/}$ is equivalently a [[continuous function]] $x \colon \ast  \to (X,\tau)$, which is equivalently just a choice of point in $X$, and a [[morphism]] in $Top^{\ast/}$ is a morphism $f \colon X \to Y$ in [[Top]] (hence a continuous function), such that this triangle [[diagram]] [[commuting diagram|commutes]]

\begin{displaymath}
\itexarray{
    && \ast
    \\
    & {}^{\mathllap{x}}\swarrow && \searrow^{\mathrlap{y}}
    &&
    \\
    X && \underset{f}{\longrightarrow} && Y
  }
\end{displaymath}
which equivalently means that $f(x) = y$.

\hypertarget{ForgettingAndAdjoiningBasepoints}{}\subsubsection*{{Forgetting and adjoining basepoints}}\label{ForgettingAndAdjoiningBasepoints}

\begin{defn}
\label{BasePointAdjoined}\hypertarget{BasePointAdjoined}{}
The [[forgetful functor]] $Top^{\ast/} \to Top$ has a [[left adjoint]] given by forming the [[disjoint union space]] ([[coproduct]] in [[Top]]) with a [[point space]] (``adjoining a base point''), this is denoted by

\begin{displaymath}
(-)_+ \coloneqq (-) \sqcup \ast \;\colon \; Top \longrightarrow Top^{\ast/}
  \,.
\end{displaymath}
\end{defn}
\hypertarget{wedge_sum_and_smash_product}{}\subsubsection*{{Wedge sum and Smash product}}\label{wedge_sum_and_smash_product}

\begin{example}
\label{WedgeSumAsCoproduct}\hypertarget{WedgeSumAsCoproduct}{}
Given two pointed topoligical spaces $(X,x)$ and $(Y,y)$, then:

\begin{enumerate}%
\item their [[Cartesian product]] in $Top^{\ast/}$ is simply their [[product topological space]] $X \times Y$ equipped with the [[pair]] of basepoints $(X\times Y, (x,y))$;


\item their [[coproduct]] in $Top^{\ast/}$ has to be computed using the second clause in \href{pointed+object#LimitsAndColimitsOfPointedObjects}{this prop.}: since the point $\ast$ has to be adjoined to the diagram, it is given not by the coproduct in $Top$ (which is the [[disjoint union space]]), but by the [[pushout]] in $Top$ of the form:

\begin{displaymath}
\itexarray{
    \ast &\overset{x}{\longrightarrow}& X
    \\
    {}^{\mathllap{y}}\downarrow &(po)& \downarrow
    \\
    Y &\longrightarrow& X \vee Y
  }
  \,.
\end{displaymath}
This is called the \emph{[[wedge sum]]} operation on pointed objects.

This is the [[quotient topological space]] of the [[disjoint union space]] under the [[equivalence relation]] which identifies the two basepoints:

\begin{displaymath}
X \vee Y \;\simeq\; (X \sqcup Y)/(x \sim y)
\end{displaymath}


\end{enumerate}
Generally for a set $\{(X_i,x_i)\}_{i \in I}$ of pointed topological spaces

\begin{enumerate}%
\item their [[product]] is formed in [[Top]], as the [[product topological space]] with the [[Tychonoff topology]], with the [[tuple]] $(x_i)_{i \in I} \in \underset{i \in I}{\prod} X_i$ of basepoints being the new basepoint;


\item their [[coproduct]] is formed by the [[colimit]] in $Top$ over the [[diagram]] with a basepoint adjoined, and is called the [[wedge sum]] $\vee_{i \in I} X_i$, which is the [[quotient topological space]] of the [[disjoint union space]] with all the basepoints identified:

\begin{displaymath}
\underset{i \in I}{\vee} X_i
  \;\simeq\;
  \left(\underset{i \in I}{\sqcup} X_i\right)/(x_i \sim x_j)_{i,j \in I}
 \,.
\end{displaymath}


\end{enumerate}
\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $X$ a [[CW-complex]], then for every $n \in \mathbb{N}$ the [[quotient]] of its $n$-skeleton by its $(n-1)$-skeleton is the [[wedge sum]], def. \ref{WedgeSumAsCoproduct}, of $n$-spheres, one for each $n$-cell of $X$:

\begin{displaymath}
X^n / X^{n-1} \simeq \underset{i \in I_n}{\vee} S^n
  \,.
\end{displaymath}
\end{example}
\begin{defn}
\label{SmashProductOfPointedObjects}\hypertarget{SmashProductOfPointedObjects}{}
The \emph{[[smash product]]} of pointed topological spaces is the [[functor]]

\begin{displaymath}
(-)\wedge(-)
    \;\colon\;
  Top^{\ast/}
    \times 
  Top^{\ast/}
    \longrightarrow
  Top^{\ast/}
\end{displaymath}
given by

\begin{displaymath}
X \wedge Y
  \;\coloneqq\;
  \ast
  \underset{X\sqcup Y}{\sqcup} (X \times Y)
  \,,
\end{displaymath}
hence by the [[pushout]] in $Top$ of he frm

\begin{displaymath}
\itexarray{
    X \sqcup Y &\overset{(id_X,y),(x,id_Y) }{\longrightarrow}& X \times Y
    \\
    \downarrow &(po)& \downarrow
    \\
    \ast &\longrightarrow& X \wedge Y
  }
  \,.
\end{displaymath}
In terms of the [[wedge sum]] from def. \ref{WedgeSumAsCoproduct}, this may be written concisely as the [[quotient space]] (\href{quotient+space#QuotientBySubspace}{this def}) of the [[product topological space]] by the [[subspace]] constituted by the [[wedge sum]]

\begin{displaymath}
X \wedge Y \simeq \frac{X\times Y}{X \vee Y}
  \,.
\end{displaymath}
\end{defn}
t $\,$

\begin{tabular}{l|l|l}
symbol&name&category theory\\
\hline 
$X \times Y$&[[product space]]&[[product]] in $Top^{\ast/}$\\
$X \vee Y$&[[wedge sum]]&[[coproduct]] in $Top^{\ast/}$\\
$X \wedge Y = \frac{X \times Y}{X \vee Y}$&[[smash product]]&[[tensor product]] in $Top^{\ast/}$\\
\end{tabular}

\begin{example}
\label{WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces}\hypertarget{WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces}{}
For $X, Y \in Top$, with $X_+,Y_+ \in Top^{\ast/}$, def. \ref{BasePointAdjoined}, then

\begin{itemize}%
\item $X_+ \vee Y_+ \simeq (X \sqcup Y)_+$;


\item $X_+ \wedge Y_+ \simeq (X \times Y)_+$.



\end{itemize}
\end{example}
\begin{proof}
By example \ref{WedgeSumAsCoproduct}, $X_+ \vee Y_+$ is given by the colimit in $Top$ over the diagram

\begin{displaymath}
\itexarray{
    && && \ast
    \\
    && & \swarrow && \searrow
    \\
    X &\,\,& \ast && && \ast &\,\,& Y
  }  
  \,.
\end{displaymath}
This is clearly $A \sqcup \ast \sqcup B$. Then, by definition \ref{SmashProductOfPointedObjects}

\begin{displaymath}
\begin{aligned}
    X_+ \wedge Y_+
    & \simeq
    \frac{(X \sqcup \ast) \times (X \sqcup \ast)}{(X\sqcup \ast) \vee (Y \sqcup \ast)}
    \\
    & \simeq 
    \frac{X \times Y \sqcup X \sqcup Y \sqcup \ast}{X \sqcup Y \sqcup \ast}
    \\
    & \simeq
    X \times Y \sqcup \ast
    \,.
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{example}
\label{StandardReducedCyclinderInTop}\hypertarget{StandardReducedCyclinderInTop}{}
Let $I \coloneqq [0,1] \subset \mathbb{R}$ be the [[closed interval]] with its [[Euclidean space|Euclidean]] [[metric topology]].

Hence

\begin{displaymath}
I_+ \in Top^{\ast/}
\end{displaymath}
is the interval with a disjoint basepoint adjoined, def. \ref{BasePointAdjoined}.

Now for $X$ any [[pointed topological space]], then the [[smash product]] (def. \ref{SmashProductOfPointedObjects})

\begin{displaymath}
X \wedge (I_+) = (X \times I)/(\{x_0\} \times I)
\end{displaymath}
is the \textbf{[[reduced cylinder]]} over $X$: the result of forming the ordinary [[cylinder]] over $X$, and then identifying the interval over the basepoint of $X$ with the point.

(Generally, any construction in $Top$ properly adapted to pointed spaces is called the ``reduced'' version of the unpointed construction. Notably so for ``[[reduced suspension]]'' which we come to \hyperlink{MappingCones}{below}.)

Just like the ordinary cylinder $X\times I$ receives a canonical injection from the [[coproduct]] $X \sqcup X$ formed in $Top$, so the reduced cyclinder receives a canonical injection from the coproduct $X \sqcup X$ formed in $Top^{\ast/}$, which is the [[wedge sum]] from example \ref{WedgeSumAsCoproduct}:

\begin{displaymath}
X \vee X \longrightarrow X \wedge (I_+)
  \,.
\end{displaymath}
\end{example}
\hypertarget{mapping_cocones}{}\subsubsection*{{Mapping (co-)cones}}\label{mapping_cocones}

Recall that the \emph{[[cone]]} on a [[topological space]] $X$ is the [[quotient space]] of the [[product space]] with the closed interval

\begin{displaymath}
Cone(X)
   = 
  (X \times [0,1])/( X \times \{0\} )
  \,.
\end{displaymath}
If $X$ is pointed with basepoint $x \in X$, then the \emph{reduced cone} is the further quotient by the copy of the interval over the basepoint

\begin{displaymath}
Cone(X,x) = Cone(X) / ( \{x\} \times [0,1] )
  \,.
\end{displaymath}
For $f \colon X \to Y$ a [[continuous function]], then

\begin{enumerate}%
\item the \emph{[[mapping cylinder]]} of $f$ is the [[attachment space]]

\begin{displaymath}
Cyl(f) \coloneqq Y \cup_f Cyl(X)
\end{displaymath}

\item the \emph{[[mapping cone]]} of $f$ is the [[attachment space]]

\begin{displaymath}
Cone(f) \coloneqq Y \cup_f Cone(X)
\end{displaymath}


\end{enumerate}
accordingly if $f \colon X \to Y$ is a continuous function between pointed spaces which preserves the basepoint, then the analogous construction with the [[reduced cylinder]] and the reduce cone, respectively, yield the \emph{reduced mapping cyclinder} and the \emph{reduced mapping cone}.

We now say this again in terms of [[pushouts]]:

\begin{defn}
\label{MappingConeAndMappingCocone}\hypertarget{MappingConeAndMappingCocone}{}
For $f \colon X \longrightarrow Y$ a [[continuous function]] between pointed spces, its \textbf{reduced [[mapping cone]]} is the space

\begin{displaymath}
Cone(f)
    \coloneqq
  \ast \underset{X}{\sqcup} Cyl(X) \underset{X}{\sqcup} Y
\end{displaymath}
in the [[colimit|colimiting]] [[diagram]]

\begin{displaymath}
\itexarray{
    && X &\stackrel{f}{\longrightarrow}& Y
    \\
    && \downarrow^{\mathrlap{i_1}} && \downarrow^{\mathrlap{i}}
    \\
    X &\stackrel{i_0}{\longrightarrow}& Cyl(X)
    \\
    \downarrow && & \searrow^{\mathrlap{\eta}} & \downarrow
    \\
    {*} &\longrightarrow& &\longrightarrow& Cone(f)
  }
  \,,
\end{displaymath}
where $Cyl(X)$ is the [[reduced cylinder]] from def. \ref{StandardReducedCyclinderInTop}.

\end{defn}
\begin{prop}
\label{ConeAndMappingCylinder}\hypertarget{ConeAndMappingCylinder}{}
The colimit appearing in the definition of the reduced [[mapping cone]] in def. \ref{MappingConeAndMappingCocone} is equivalent to three consecutive [[pushouts]]:

\begin{displaymath}
\itexarray{
    && X &\stackrel{f}{\longrightarrow}& Y
    \\
    && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{i}}
    \\
    X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cyl(f)
    \\
    \downarrow &(po)& \downarrow & (po) & \downarrow
    \\
    {*} &\longrightarrow& Cone(X) &\longrightarrow& Cone(f)
  }
  \,.
\end{displaymath}
The two intermediate objects appearing here are called

\begin{itemize}%
\item the plain \textbf{reduced [[cone]]} $Cone(X) \coloneqq \ast \underset{X}{\sqcup} Cyl(X)$;


\item the \textbf{reduced [[mapping cylinder]]} $Cyl(f) \coloneqq Cyl(X) \underset{X}{\sqcup} Y$.



\end{itemize}
\end{prop}
\begin{defn}
\label{SuspensionAndLoopSpaceObject}\hypertarget{SuspensionAndLoopSpaceObject}{}
Let $X \in Top^{\ast/}$ be any pointed topological space.

The [[mapping cone]], def. \ref{ConeAndMappingCylinder}, of $X \to \ast$ is called the \textbf{[[reduced suspension|reduced]] [[suspension]]} of $X$, denoted

\begin{displaymath}
\Sigma X = Cone(X\to\ast)\,.
\end{displaymath}
Via prop. \ref{ConeAndMappingCylinder} this is equivalently the coproduct of two copies of the cone on $X$ over their base:

\begin{displaymath}
\itexarray{
       && X &\stackrel{}{\longrightarrow}& \ast
       \\
       && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{}}
       \\
       X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cone(X)
       \\
       \downarrow &(po)& \downarrow & (po) & \downarrow
       \\
       {*} &\longrightarrow& Cone(X) &\longrightarrow& \Sigma X
     }
     \,.
\end{displaymath}
This is also equivalently the [[cofiber]]f of $(i_0,i_1)$, hence (example \ref{WedgeSumAsCoproduct}) of the [[wedge sum]] inclusion:

\begin{displaymath}
X \vee X
      =
     X \sqcup X
      \overset{(i_0,i_1)}{\longrightarrow}
     Cyl(X)
       \overset{cofib(i_0,i_1)}{\longrightarrow}
     \Sigma X
     \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{ReducedSuspensionBySmashProductWithCircle}\hypertarget{ReducedSuspensionBySmashProductWithCircle}{}
The [[reduced suspension]] objects (def. \ref{SuspensionAndLoopSpaceObject}) induced from the standard [[reduced cylinder]] $(-)\wedge (I_+)$ of example \ref{StandardReducedCyclinderInTop} are isomorphic to the [[smash product]] (def. \ref{SmashProductOfPointedObjects}) with the [[circle] (the [[1-sphere]])

\begin{displaymath}
cofib(X \vee X \to X \wedge (I_+)) \simeq S^1 \wedge X
    \,,
\end{displaymath}
\end{prop}
\begin{prop}
\label{UnreducedMappingConeAsReducedConeOfBasedPointAdjoined}\hypertarget{UnreducedMappingConeAsReducedConeOfBasedPointAdjoined}{}
For $f \colon X \longrightarrow Y$ a morphism in [[Top]], then its unreduced mapping cone with respect to the standard cylinder object $X \times I$ def. \ref{TopologicalInterval}, is isomorphic to the reduced mapping cone, of the morphism $f_+ \colon X_+ \to Y_+$ (with a basepoint adjoined) with respect to the standard [[reduced cylinder]]:

\begin{displaymath}
Cone'(f) \simeq Cone(f_+)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By example \ref{WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces}, $Cone(f_+)$ is given by the colimit in $Top$ over the following diagram:

\begin{displaymath}
\itexarray{
    \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast
    \\
    \downarrow && && \downarrow
    \\
    \ast &\longrightarrow& &\longrightarrow& Cone(f_+)
  }
  \,.
\end{displaymath}
We may factor the vertical maps to give

\begin{displaymath}
\itexarray{
    \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast
    \\
    \downarrow && && \downarrow
    \\
    \ast \sqcup \ast &\longrightarrow& &\longrightarrow& Cone'(f)_+
    \\
    \downarrow && && \downarrow
    \\
    \ast &\longrightarrow& &\longrightarrow& Cone'(f)
  }
  \,.
\end{displaymath}
This way the top part of the diagram (using the [[pasting law]] to compute the colimit in two stages) is manifestly a cocone under the result of applying $(-)_+$ to the diagram for the unreduced cone. Since $(-)_+$ is itself given by a colimit, it preserves colimits, and hence gives the partial colimit $Cone'(f)_+$ as shown. The remaining pushout then contracts the remaining copy of the point away.

\end{proof}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

Most of the relevant constructions on pointed topological spaces are immediate specializations of the general construction discussed at \emph{[[pointed object]]}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[pointed simplicial set]]


\item [[loop space]], [[reduced suspension]]


\item [[smash product]], [[wedge sum]]


\item [[pointed homotopy type]]


\item [[spectrum]]


\item [[classical model structure on pointed topological spaces]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Pierre Gabriel]], [[Michel Zisman]], chapter V.7 of \emph{[[Calculus of fractions and homotopy theory]]}, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) (\href{https://www.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf}{pdf})

\end{itemize}
[[!redirects pointed topological spaces]]

[[!redirects based topological space]] [[!redirects based topological spaces]]



\end{document}