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\begin{document}

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\section*{topological cofiber sequence}

\begin{quote}%
under construction


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

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

[[!include topology - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{background}{Background}\dotfill \pageref*{background} \linebreak
\noindent\hyperlink{homotopy_equivalences}{Homotopy equivalences}\dotfill \pageref*{homotopy_equivalences} \linebreak
\noindent\hyperlink{cofibrations}{Cofibrations}\dotfill \pageref*{cofibrations} \linebreak
\noindent\hyperlink{pushouts__space_attachments}{Pushouts / space attachments}\dotfill \pageref*{pushouts__space_attachments} \linebreak
\noindent\hyperlink{mapping_cones}{Mapping cones}\dotfill \pageref*{mapping_cones} \linebreak
\noindent\hyperlink{CofiberSequences}{Cofiber sequences}\dotfill \pageref*{CofiberSequences} \linebreak
\noindent\hyperlink{pointed_cofiber_sequence}{Pointed cofiber sequence}\dotfill \pageref*{pointed_cofiber_sequence} \linebreak


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

We discuss in detail the realization of the abstract concept of [[cofiber sequences]] in its explicit incarnation in [[point-set topology]], the way it is traditionally presented in [[topology]] textbooks.

Hence we use the concepts of [[homotopy equivalence]] instead of [[weak homotopy equivalence]]. For discussion using the latter in the context of the [[classical model structure on topological spaces]] see instead at \emph{[[Introduction to Homotopy Theory]]} the section \emph{\href{Introduction+to+Homotopy+Theory#HomotopyFiberSequences}{Homotopy fiber sequences}}.

\hypertarget{background}{}\subsection*{{Background}}\label{background}

Throughout, write $I \coloneqq [0,1] \subset \mathbb{R}$ for the [[closed interval]] equipped with its [[Euclidean space|Euclidean]] [[metric topology]].

\hypertarget{homotopy_equivalences}{}\subsubsection*{{Homotopy equivalences}}\label{homotopy_equivalences}

\begin{defn}
\label{LeftHomotopy}\hypertarget{LeftHomotopy}{}
For $f,g\colon X \longrightarrow Y$ two [[continuous functions]] between [[topological spaces]] $X,Y$, then a \textbf{[[homotopy]]}

\begin{displaymath}
\eta \colon f \,\Rightarrow_L\, g
\end{displaymath}
is a [[continuous function]]

\begin{displaymath}
\eta \;\colon\; X \times I \longrightarrow Y
\end{displaymath}
out of the standard [[cylinder object]] over $X$: the [[product space]] of $X$ with the [[Euclidean space|Euclidean]] [[closed interval]], such that this fits into a [[commuting diagram]] of the form

\begin{displaymath}
\itexarray{
     X
     \\
     {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}}
     \\
     X \times I &\stackrel{\eta}{\longrightarrow}& Y
     \\
     {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}}
     \\
     X
  }
  \,.
\end{displaymath}
(graphics grabbed from J. Tauber \href{http://jtauber.com/blog/2005/07/01/path_homotopy/}{here})

\end{defn}
\begin{defn}
\label{HomotopyEquivalence}\hypertarget{HomotopyEquivalence}{}
\textbf{([[homotopy equivalence]])}

A [[continuous function]] $f \colon X \to Y$ is called a \emph{[[homotopy equivalence]]} if there exist

\begin{enumerate}%
\item a continuous function $g \colon Y \to X$;


\item homotopies (def. \ref{LeftHomotopy}) of the form

$g \circ f \Rightarrow id_X \phantom{AAAA} f \circ g \Rightarrow id_Y$



\end{enumerate}
\end{defn}
\begin{example}
\label{ContractibleTopologicalSpace}\hypertarget{ContractibleTopologicalSpace}{}
\textbf{([[contractible topological space]])}

A [[topological space]] $X$ is called \emph{[[contractible topological space|contractible]]} if the unique map to the [[point space]] $X \to \ast$ is a [[homotopy equivalence]] (def. \ref{HomotopyEquivalence}).

\end{example}
\hypertarget{cofibrations}{}\subsubsection*{{Cofibrations}}\label{cofibrations}

\begin{defn}
\label{HurewiczCofibration}\hypertarget{HurewiczCofibration}{}
\textbf{([[Hurewicz cofibration]])}

A [[continuous function]] $i \colon A \longrightarrow X$ is called a \emph{[[Hurewicz cofibration]]} or just \emph{cofibration} if it satisfies the [[homotopy extension property]] in that:

\begin{itemize}%
\item for any [[topological space]] $Y$,


\item for all continuous functions $f \colon A\to Y$, $\tilde{f}:X\to Y$ such that $\tilde{f}\circ i=f$

\begin{displaymath}
\itexarray{
     A &\stackrel{f}{\to}& Y
     \\
     {}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\tilde f}}
     \\
     X
  }
\end{displaymath}

\item and any [[left homotopy]] $F \colon A\times I\to Y$ such that $F(-,0)=f$



\end{itemize}
there is a homotopy $\tilde{F} \colon X\times I\to Y$ such that

\begin{itemize}%
\item $\tilde{F}\circ(i\times id_I)=F$

\begin{displaymath}
\itexarray{
     A \times I &\overset{F}{\longrightarrow}& Y
     \\
     {}^{\mathllap{i \times id_I}}\downarrow & \nearrow_{\mathrlap{\tilde F}}
     \\
     X \times I
  }
\end{displaymath}

\item and $\tilde{F}(-,0)=\tilde{f}$

\begin{displaymath}
\itexarray{
     A &\overset{id_A \times const_0}{\longrightarrow}& A \times I &\overset{F}{\longrightarrow}& Y
     \\
     {}^{\mathllap{i \times id_I}}\downarrow
     && {}^{\mathllap{i \times id_I}}\downarrow & \nearrow_{\mathrlap{\tilde F}}
     \\
     X &\underset{id_X \times const_0}{\longrightarrow}& X \times I
  }
  \phantom{AAA}
    =
  \phantom{AAA}
  \itexarray{
  \itexarray{
     A &\stackrel{f}{\to}& Y
     \\
     {}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\tilde f}}
     \\
     X
  }cco
  }
\end{displaymath}


\end{itemize}
If the [[image]] $i(A) \subset X$ is a [[closed subset]], then $i$ is called a \emph{[[closed Hurewicz cofibration]]}.

\end{defn}
\begin{example}
\label{RelativeCellComplexInclusionsAreClosedCofibrations}\hypertarget{RelativeCellComplexInclusionsAreClosedCofibrations}{}
\textbf{([[retracts]] of [[relative cell complex]] inclusions are [[closed Hurewicz cofibrations]])}

If $A$ is a [[topological space]] and $i \colon A \hookrightarrow X$ is a [[relative cell complex]] inclusion, then $i$ is a [[closed Hurewicz cofibration]] (def. \ref{HurewiczCofibration}).

Also every [[retract]] of such $i$ (\href{Introduction+to+Homotopy+Theory#RetractsOfMorphisms}{this sense}) is a closed Hurewicz cofibration.

\end{example}
\hypertarget{pushouts__space_attachments}{}\subsubsection*{{Pushouts / space attachments}}\label{pushouts__space_attachments}

Cofiber sequences (\hyperlink{CofiberSequences}{below}) are constructed by iterated \emph{[[pushouts]]} in the [[category]] [[Top]] of [[topological spaces]] with [[continuous functions]] between them ([[space attachments]]), see at \emph{\href{Top#UniversalConstructions}{Top -- Universal constructions}}.

\begin{example}
\label{QuotientBySubspace}\hypertarget{QuotientBySubspace}{}
\textbf{([[quotient space]] by a [[subspace]])}

Let $X$ be a [[topological space]] and $A \subset X$ a [[inhabited set|non-empty]] [[subset]]. Consider the [[equivalence relation]] on $X$ which identifies all points in $A$ with each other. The resulting [[quotient space]] is denoted $X/A$.

Notice that $X/A$ is canonically a [[pointed topological space]], with base point the [[equivalence class]] $A/A \subset X/A$ of $A$.

If $A = \emptyset$ is the [[empty space]], then one defines

\begin{displaymath}
X/\emptyset \coloneqq X_+ \coloneqq X \sqcup \ast
\end{displaymath}
to be the [[disjoint union space]] of $X$ with the [[point space]]. This is no longer a quotient space, but both constructions are unified by the \emph{[[pushout]]} $i \colon A \to X$ along the map $A \to \ast$, equivalently the [[cokernel]] of the inclusion:

\begin{displaymath}
\itexarray{
    A &\overset{i}{\hookrightarrow}& X
    \\
    \downarrow &(po)& \downarrow
    \\
    \ast &\longrightarrow& X/A
  }
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{ClosedSubspacesGluing}\hypertarget{ClosedSubspacesGluing}{}
\textbf{([[union]] of two [[open subset|open]] or two [[closed subset|closed]] [[subspaces]] is [[pushout]])}

Let $X$ be a [[topological space]] and let $A,B \subset X$ be [[subspaces]] such that

\begin{enumerate}%
\item $A,B \subset X$ are both [[open subsets]] or are both [[closed subsets]];


\item they constitute a [[cover]]: $X = A \cup B$



\end{enumerate}
Write $i_A \colon A \to X$ and $i_B \colon B \to X$ for the corresponding inclusion [[continuous functions]].

Then the [[commuting square]]

\begin{displaymath}
\itexarray{
     A \cap B &\longrightarrow& A
     \\
     \downarrow &(po)& \downarrow^{\mathrlap{i_A}}
     \\
     B &\underset{i_B}{\longrightarrow}& X
  }
\end{displaymath}
is a [[pushout]] square in [[Top]] (see \href{Top#UniversalConstructions}{there}).

By the [[universal property]] of the [[pushout]] this means in particular that for $Y$ any [[topological space]] then a function of underlying sets

\begin{displaymath}
f \;\colon\; X \longrightarrow Y
\end{displaymath}
is a [[continuous function]] as soon as its two restrictions

\begin{displaymath}
f\vert_A \;\colon\; A \longrightarrow Y
  \phantom{AAAA}
  f\vert_A \;\colon\; B \longrightarrow Y
\end{displaymath}
are continuous.

\end{example}
\begin{proof}
Clearly the underlying diagram of underlying [[sets]] is a pushout in [[Set]]. Therefore (by \href{Top#DescriptionOfLimitsAndColimitsInTop}{this prop.}) we need to show that the [[topological space|topology]] on $X$ is the [[final topology]] induced by the set of functions $\{i_A, i_B\}$, hence that a [[subset]] $S \subset X$ is an [[open subset]] precisely if the [[pre-images]] (restrictions)

\begin{displaymath}
i_A^{-1}(S) = S \cap A
   \phantom{AAA}
    \text{and}
   \phantom{AAA}
  i_B^{-1}(S) = S \cap B
\end{displaymath}
are open subsets of $A$ and $B$, respectively.

Now by definition of the [[subspace topology]], if $S \subset X$ is open, then the intersections $A \cap S \subset A$ and $B \cap S \subset B$ are open in these subspaces.

Conversely, assume that $A \cap S \subset A$ and $B \cap S \subset B$ are open. We need to show that then $S \subset X$ is open.

Consider now first the case that $A;B \subset X$ are both open open. Then by the nature of the [[subspace topology]], that $A \cap S$ is open in $A$ means that there is an open subset $S_A \subset X$ such that $A \cap S = A \cap S_A$. Since the intersection of two open subsets is open, this implies that $A \cap S_A$ and hence $A \cap S$ is open. Similarly $B \cap S$. Therefore

\begin{displaymath}
\begin{aligned}
    S
     & =
    S \cap X
    \\
    & = S \cap (A \cup B)
    \\
    & =  (S \cap A) \cup (S \cap B)
  \end{aligned}
\end{displaymath}
is the union of two open subsets and therefore open.

Now consider the case that $A,B \subset X$ are both closed subsets.

Again by the nature of the subspace topology, that $A \cap S \subset A$ and $B \cap S \subset B$ are open means that there exist open subsets $S_A, S_B \subset X$ such that $A \cap S = A \cap S_A$ and $B \cap S = B \cap S_B$. Since $A,B \subset X$ are closed by assumption, this means that $A \setminus S, B \setminus S \subset X$ are still closed, hence that $X \setminus (A \setminus S), X \setminus (B \setminus S) \subset X$ are open.

Now observe that (by [[de Morgan duality]])

\begin{displaymath}
\begin{aligned}
    S
      & = X \setminus (X \setminus S)
    \\
      & =
    X \setminus ( (A \cup B) \setminus S )
    \\
      & =
    X \setminus ( (A \setminus S) \cup (B \setminus S) )
    \\
     & =
    (X \setminus (A \setminus S)) \cap (X \setminus (B \setminus S))
    \,.
  \end{aligned}
\end{displaymath}
This exhibits $S$ as the intersection of two open subsets, hence as open.

\end{proof}
A general abstract fact about pushouts which we will use repeatedly in the proofs below is the following:

\begin{prop}
\label{PastingLaw}\hypertarget{PastingLaw}{}
\textbf{([[pasting law]])}

Consider a [[diagram]] in [[Top]] (or in any other [[category]]) of the following form:

\begin{displaymath}
\itexarray{
     &\longrightarrow& &\longrightarrow&
    \\
    \downarrow &(po)& \downarrow && \downarrow
    \\
     &\longrightarrow& &\longrightarrow&
  }
 \,,
\end{displaymath}
where the left square is a [[pushout]].

Then: The total rectangle is a [[pushout]] precisely if the right square is.

\end{prop}
We need the following two facts (prop. \ref{CofibrantHomotopyEquivalencePushout} prop. \ref{ContractibleSubcomplexQuotient}) regarding the stability of cofibrations under pushout.

\begin{prop}
\label{CofibrantHomotopyEquivalencePushout}\hypertarget{CofibrantHomotopyEquivalencePushout}{}
\textbf{([[pushout]] of cofibrations)}

Let $A$ be a [[topological space]] and let $A \hookrightarrow X$ be a [[closed Hurewicz cofibration]].

Then for every [[continuous function]] $f \colon A \to Y$, the [[pushout]] $f_\ast i$ in

\begin{displaymath}
\itexarray{
     A &\overset{f}{\longrightarrow}& Y
     \\
     {}^{\mathllap{i}}\downarrow &(po)& \downarrow^{\mathrlap{f_\ast i}}
     \\
     X &\longrightarrow& X \underset{A}{\sqcup} Y
  }
\end{displaymath}
is also a [[closed Hurewicz cofibration]]. Moreover, if in addition

\begin{enumerate}%
\item $i$ is a [[retract]] of a [[relative cell complex]] inclusion, then so is $f_\ast i$,


\item $i$ is a [[homotopy equivalence]], then so is $f_\ast i$.



\end{enumerate}
\end{prop}
Prop. \ref{CofibrantHomotopyEquivalencePushout} is a consequence of the existence of the [[Strøm model structure]] $Top_{Strom}$ and the [[classical model structure on topological spaces]] $Top_{Quillen}$ and of the fact that the identity [[functors]] $Top_{Strom} \underoverset{\underset{id}{\longrightarrow}}{\overset{id}{\longleftarrow}}{\bot} Top_{Quillen}$ constitute a [[Quillen adjunction]].

\begin{prop}
\label{ContractibleSubcomplexQuotient}\hypertarget{ContractibleSubcomplexQuotient}{}
\textbf{([[quotient space|quotient]] by [[contractible topological space|contractible]] closed subspace)}

Let $i \colon A \longrightarrow X$ be a [[closed Hurewicz cofibration]].

If $f \colon A \to Y$ is a [[homotopy equivalence]] (def. \ref{HomotopyEquivalence}) then its [[pushout]] $i_\ast f$ in

\begin{displaymath}
\itexarray{
     A &\overset{i}{\longrightarrow}& X
     \\
     {}^{\mathllap{f}}\downarrow &(po)& \downarrow^{\mathrlap{i_\ast(f)}}
     \\
     Y &\longrightarrow& Y \underset{A}{\sqcup} X
  }
\end{displaymath}
is also a homotopy equivalence.

In particular if $A$ is a [[contractible topological space]] (example \ref{ContractibleTopologicalSpace}), then the coprojection

\begin{displaymath}
X \longrightarrow X/A
\end{displaymath}
to the [[quotient space]] (example \ref{QuotientBySubspace}) is a [[homotopy equivalence]].

\end{prop}
Prop. \ref{ContractibleSubcomplexQuotient} is again a consequence of the existence of the [[Strøm model structure]] $Top_{Strom}$: It is the statement that $Top_{Strom}$ is a [[left proper model category]] which follows (\href{proper%20model%20category#AllObjectsFibrantImpliesRightProper}{this cor.}) since all its objects are evidently cofibrant. An elementary proof in [[point-set topology]] is offered in (\href{algebraic+topology#Hatcher}{Hatcher, prop. 0.17})

\hypertarget{mapping_cones}{}\subsection*{{Mapping cones}}\label{mapping_cones}

\begin{defn}
\label{ConeCyclinder}\hypertarget{ConeCyclinder}{}
\textbf{(topological [[cylinder]] and cone)}

Let $X$ be a [[topological space]]. Then

\begin{enumerate}%
\item the \emph{standard [[cylinder]]} on $X$ is the [[product topological space]]

\begin{displaymath}
Cyl(X) \coloneqq X \times [0,1]
\end{displaymath}

\item the \emph{standard [[cone]]} on $X$ is the [[quotient space]] (example \ref{QuotientBySubspace})

\begin{displaymath}
Cone(X) \coloneqq Cyl(X) / (X \times \{0\})
\end{displaymath}
of the standard cylinder by the [[subspace]] $X \times \{0\} \subset X \times [0,1]$.

Equivalently this is the following [[pushout]] in [[Top]]

\begin{displaymath}
\itexarray{
     X \times \{0\} &\hookrightarrow& Cyl(X)
     \\
     \downarrow &(po)& \downarrow
     \\
     \ast &\longrightarrow& Cone(X)
  }
\end{displaymath}


\end{enumerate}
\end{defn}
\begin{lemma}
\label{InclusionIntoCylinderIsHomotopyEquivalence}\hypertarget{InclusionIntoCylinderIsHomotopyEquivalence}{}
Let $X$ be a topological space. Then the canonical inclusions

\begin{displaymath}
X \longrightarrow Cyl(X)
  \phantom{AAAA}
  \ast \longrightarrow Cone(X)
\end{displaymath}
are [[homotopy equivalences]]. Hence every [[cone]] is a contractible topological space] (example \ref{ContractibleTopologicalSpace}).

\end{lemma}
\begin{defn}
\label{MappingConeCylinder}\hypertarget{MappingConeCylinder}{}
\textbf{([[mapping cylinder]] and [[mapping cone]])}

Let $f \colon X \to Y$ be a [[continuous function]] between [[topological spaces]]. Then

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

\begin{displaymath}
Cyl(f) \coloneqq Y \cup_f Cyl(X)
\end{displaymath}
of $Y$ with the [[cylinder]] on $X$, according to def. \ref{ConeCyclinder}, hence the following [[pushout]] in [[Top]]

\begin{displaymath}
\itexarray{
    X &\overset{f}{\longrightarrow}& Y
    \\
    \downarrow &(po)& \downarrow
    \\
    Cyl(X) &\longrightarrow& Cyl(f)
  }
\end{displaymath}

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

\begin{displaymath}
Cone(f) \coloneqq Y \cup_f Cone(X)
\end{displaymath}
hence the following [[pushout]] in [[Top]]:

\begin{displaymath}
\itexarray{
    X &\overset{f}{\longrightarrow}& Y
    \\
    \downarrow &(po)& \downarrow
    \\
    Cone(X) &\longrightarrow& Cone(f)
  }
\end{displaymath}


\end{enumerate}
\end{defn}
\begin{remark}
\label{MappingConePasting}\hypertarget{MappingConePasting}{}
In summary, def. \ref{ConeCyclinder} and def. \ref{MappingConeCylinder} say that for $f \colon X \to Y$ a [[continuous function]] then we have a [[pasting]] of [[pushout]] diagrams in [[Top]] of the following form:

\begin{displaymath}
\itexarray{
    && X &\stackrel{f}{\to}& Y
    \\
    && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow
    \\
    X &\stackrel{i_0}{\to}& Cyl(X) &\to & Cyl(f)
    \\
    \downarrow &(po)& \downarrow &(po)& \downarrow
    \\
    {*} &\to& Cone(X) &\to& Cone(f)
  }
  \,.
\end{displaymath}
Since $X \to Cone(X)$ is a [[closed Hurewicz cofibration]] , the [[pasting law]] together with prop. \ref{CofibrantHomotopyEquivalencePushout} therefore implies that also $Y \to Cone(f)$ is a closed Hurwicz cofibration.

\end{remark}
\begin{example}
\label{Suspension}\hypertarget{Suspension}{}
\textbf{([[suspension]])}

For $X$ a [[topological space]], then the [[mapping cone]] (def. \ref{MappingConeCylinder}) of the unique function $X \to \ast$ to the [[point space]] is

\begin{displaymath}
S X
    \coloneqq
  Cone(X \to \ast)
   \simeq
  Cone(X) \underset{X}{\sqcup} Cone(X)
  \,.
\end{displaymath}
This is called the \emph{[[suspension]]} of $X$.

\end{example}
The mapping cone of a map $X \to Y$ is to be thought of as the homotopy-quotient of the $Y$ by $X$, as opposed to be the naive quotient. This is made precise by the following two statements, lemma \ref{RelativeCellComplexInclusionMappingCone} and lemma \ref{ForClosedImageQuotientOfMappingConeByCone}:

\begin{lemma}
\label{RelativeCellComplexInclusionMappingCone}\hypertarget{RelativeCellComplexInclusionMappingCone}{}
If $f \colon X \to Y$ is a [[closed Hurewicz cofibration]] (def. \ref{HurewiczCofibration}), then the coprojection

\begin{displaymath}
Cone(f) \longrightarrow Y/f(X)
\end{displaymath}
from its [[mapping cone]] (def. \ref{MappingConeCylinder}) to the naive [[quotient space]] (example \ref{QuotientBySubspace}) is a [[homotopy equivalence]].

\end{lemma}
\begin{proof}
Consider the diagram

\begin{displaymath}
\itexarray{
     X &\overset{f}{\longrightarrow}& Y
     \\
     \downarrow &(po)& \downarrow
     \\
     Cone(X) &\longrightarrow& Cone(f)
     \\
     \downarrow &(po)& \downarrow
     \\
     \ast &\longrightarrow& Cone(f)/Cone(X)
  }
\end{displaymath}
Since $f$ is a [[closed Hurewicz cofibration]], so is $Cone(X) \to Cone(f)$, by prop. \ref{CofibrantHomotopyEquivalencePushout}. Since $Cone(X) \to \ast$ is a [[homotopy equivalence]] (lemma \ref{InclusionIntoCylinderIsHomotopyEquivalence}), so is $Cone(f) \to Cone(f)/Cone(X)$, by prop. \ref{ContractibleSubcomplexQuotient}. But by lemma \ref{ForClosedImageQuotientOfMappingConeByCone} there is also a homeomorphism $Cone(f)/Cone(X) \simeq Y/f(X)$.

\end{proof}
\begin{lemma}
\label{ForClosedImageQuotientOfMappingConeByCone}\hypertarget{ForClosedImageQuotientOfMappingConeByCone}{}
Let $f \colon X \to Y$ be a [[continuous function]] such that the [[image]] $f(X) \subset Y$ is a [[closed subset]].

Then there is a [[homeomorphism]]

\begin{displaymath}
Cone(f)/Cone(X) \simeq Y / f(X)
\end{displaymath}
between the [[quotient space]] (example \ref{QuotientBySubspace}) of the [[mapping cone]] of $f$ (def. \ref{MappingConeCylinder}) by the [[cone]] of $X$ (def. \ref{ConeCyclinder}) and the [[quotient space]] of $Y$ be the [[image]] of $X$.

\end{lemma}
\begin{proof}
Consider the following [[diagram]] in [[Top]]:

\begin{displaymath}
\itexarray{
     f(X) &\longrightarrow& Y
     \\
     \downarrow &(po)& \downarrow
     \\
     f(Cone(X)) &\longrightarrow& Cone(f)
     \\
     \downarrow &(po)& \downarrow
     \\
     \ast &\longrightarrow& Cone(f)/Cone(X)
  }
\end{displaymath}
Here the top square is a [[pushout]] by example \ref{ClosedSubspacesGluing}, while the bottom square is a pushout by def. \ref{MappingConeCylinder}. Hence the total rectangle is also a pushout, by the [[pasting law]] (prop. \ref{PastingLaw}). But that total rectangle is the defining pushout for $Y/f(X)$ (by example \ref{QuotientBySubspace}). Hence the statement follows by the [[universal property]] of the pushout.

\end{proof}
\hypertarget{CofiberSequences}{}\subsection*{{Cofiber sequences}}\label{CofiberSequences}

\begin{prop}
\label{HomotopyEquivalenceSuspensionWithMappingConeOfMappingCone}\hypertarget{HomotopyEquivalenceSuspensionWithMappingConeOfMappingCone}{}
Let $f \colon X \to Y$ be a [[continuous function]] with [[closed subset|closed]] [[image]] $f(X) \subset Y$. Write $Cone(g)$ in

\begin{displaymath}
\itexarray{
    X
      &\overset{f}{\longrightarrow}&
    Y
      &\overset{g}{\longrightarrow}&
    Cone(f)
      &\overset{}{\longrightarrow}&
    Cone(g)
  }
\end{displaymath}
for the [[mapping cone]] (spring) of the inclusion $g$ of $Y$ into the mapping cone of $f$.

Then the canonical quotient coprojection

\begin{displaymath}
Cone(g) \to \Sigma X
\end{displaymath}
to the [[suspension]] of $X$ (example \ref{Suspension}) is a [[homotopy equivalence]] (def. \ref{HomotopyEquivalence}).

\end{prop}
\begin{proof}
Since $g \colon Y \to Cone(f)$ is a [[closed Hurewicz cofibration]] (by remark \ref{MappingConePasting}), lemma \ref{RelativeCellComplexInclusionMappingCone} gives that

\begin{displaymath}
Cone(g) \to Cone(f)/g(Y)
\end{displaymath}
is a [[homotopy equivalence]]. But then there is the following evident [[homeomorphism]]

\begin{displaymath}
Cone(f)/g(Y)
   =
  (Y \cup_f Cone(X))/Y
   \simeq
  Cone(X)/X
   \simeq
  S X
  \,.
\end{displaymath}
\end{proof}


(graphics taken from \href{http://personal.us.es/fmuro/praha.pdf}{Muro 10})

Hence from every $f \colon X \to Y$ with closed image, we get long sequences

\begin{displaymath}
\itexarray{
    X
      &\overset{f}{\longrightarrow}&
    Y
      &\overset{g}{\longrightarrow}&
    Cone(f)
      &\longrightarrow&
    Cone(g)
    \\
    && && &&
    \downarrow^{\mathrlap{\text{homotopy} \atop \text{equivalence}}}
    \\
    && && &&
    S X
      &\overset{S f}{\longrightarrow}&
    S Y
      &\longrightarrow& \cdots
  }
\end{displaymath}
\hypertarget{pointed_cofiber_sequence}{}\subsection*{{Pointed cofiber sequence}}\label{pointed_cofiber_sequence}

(\ldots{})



\end{document}