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\begin{document}

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\section*{topological interval}

\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{induced_constructions}{Induced constructions}\dotfill \pageref*{induced_constructions} \linebreak
\noindent\hyperlink{topological_cylinder}{Topological cylinder}\dotfill \pageref*{topological_cylinder} \linebreak
\noindent\hyperlink{left_homotopy}{Left homotopy}\dotfill \pageref*{left_homotopy} \linebreak
\noindent\hyperlink{path_space}{Path space}\dotfill \pageref*{path_space} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{freyds_characterization}{Freyd's characterization}\dotfill \pageref*{freyds_characterization} \linebreak


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

Generally, a \emph{topological interval} is a ([[bounded set|bounded]]) [[interval]] in the [[real line]] (an [[open interval]] $(a,b)$ or a [[closed interval]] $[a,b]$ or a [[half-open interval]] $(a,b]$ or $[a,b)$) equipped with the [[subspace topology]] of the [[Euclidean space|Euclidean]] [[metric topology]].

Specifically in the context of [[topological homotopy theory]], the \emph{standard topological [[interval object]]} is the [[closed interval]] $[0,1]$ equipped with the [[continuous functions]]

\begin{enumerate}%
\item $const_0 \;\colon\; \ast \to [0,1]$


\item $const_1 \;\colon\; \ast \to [0,1]$



\end{enumerate}
which include the [[point space]] as the two endpoints, respectively.

Together with the unique function $[0,1] \to \ast$ this yields the factorization of the [[codiagonal]] on the [[point space]]

\begin{displaymath}
\nabla_{\ast}
  \;\colon\;
  \ast \sqcup \ast \overset{(const_0,const_1)}{\longrightarrow} [0,1] \overset{\exists!}{\longrightarrow} \ast
\end{displaymath}
which exhibits an example of an \emph{[[interval object]]} in the general sense of [[model category]] theory with respect to the [[classical model structure on topological spaces]].

\hypertarget{induced_constructions}{}\subsection*{{Induced constructions}}\label{induced_constructions}

\hypertarget{topological_cylinder}{}\subsubsection*{{Topological cylinder}}\label{topological_cylinder}

For $X$ a [[topological space]], then the [[product topological space]] $X \times [0,1]$ with the topological interval is the \emph{standard topological [[cylinder]]} over $X$. Via the above inclusion functions, this inherits a factorization of the [[codiagonal]] of $X$ (the canonical continuous function out of the [[disjoint union space]] of $X$ with itself to $X$):

\begin{displaymath}
\nabla_X
  \;\colon\;
  X \sqcup X
   \overset{(id_X \times const_0, id_X \times const_1)}{\longrightarrow}
  X \times [0,1]
   \overet{pr_1}{\longrightarrow}
  X
  \,.
\end{displaymath}
Accordingly, with respect to the [[classical model structure on topological spaces]] this is an example a of \emph{[[cylinder object]]}.

\hypertarget{left_homotopy}{}\subsubsection*{{Left homotopy}}\label{left_homotopy}

Given $X,Y$ two [[topological spaces]] and $f,g \;\colon\; X \longrightarrow Y$ two [[continuous functions]],then a \textbf{[[left 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 [[product topological space]] of $X$ with the topological 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})

\hypertarget{path_space}{}\subsubsection*{{Path space}}\label{path_space}

For $X$ a [[topological space]], then its \emph{path space} is the [[mapping space]] $X^{[0,1]}$, out of the topological interval into $X$, i.e. the set of [[continuous function]] $\gamma \;\colon\; [0,1] \to X$ equipped with the [[compact-open topology]].

The two endpoint inclusions $\ast \colon [0,1]$ and the unique projection $[0,1] \to \ast$ induce continuous functions

\begin{displaymath}
X
    \overset{}{\longrightarrow}
  X^{[0,1]}
   \overset{X^{(const_0,const_1)}}{\longrightarrow}
  X \times X
\end{displaymath}
(inclusion of constant paths and endpoint evaluation of paths).

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

\hypertarget{freyds_characterization}{}\subsubsection*{{Freyd's characterization}}\label{freyds_characterization}

The topological interval $[0, 1]$ may be characterized by a [[coalgebra for an endofunctor|coalgebraic]] definition first identified by [[Freyd]]:

Let $Top_{\ast, \ast}$ be the [[category]] of [[topological spaces]] $X$ equipped with a [[pair]] $x_0, x_1$ of distinct points, for example $I = ([0, 1]; 0, 1)$. Let $F \colon Top_{\ast, \ast} \to Top_{\ast, \ast}$ be the [[functor]] defined on [[objects]] by

\begin{displaymath}
F(X; x_0, x_1) = (X \vee X, y_0, y_1)
  \,,
\end{displaymath}
where $X \vee X$ denotes the [[quotient space]] of the [[disjoint union space]] $X \sqcup X$ formed by identifying the element $x_1$ of the first copy of $X$ with $x_0$ of the second copy of $X$, where $y_0$ is identified with $x_0$ of the first copy, and $y_1$ is identified with $x_1$ of the second copy.

For $I = ([0, 1]; 0, 1)$ there is an evident identification $F(I) = ([0, 2]; 0, 2)$, and moreover there is an $F$-[[coalgebra of an endofunctor|coalgebra]] structure $I \to F(I)$ given by ``multiplication by $2$''.

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Freyd)}

The topological interval $I = ([0, 1], 0, 1)$ is the [[terminal coalgebra of an endofunctor|terminal F-coalgebra]].

\end{theorem}
For more information, see [[coalgebra of the real interval]], which shows in particular how the interval structure of $[0, 1]$ may be defined by coinduction.

[[!redirects topological intervals]]



\end{document}