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\begin{document}

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\section*{product topological space}

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

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

[[!include topology - contents]]

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{basic_properties}{Basic properties}\dotfill \pageref*{basic_properties} \linebreak
\noindent\hyperlink{universal_property}{Universal property}\dotfill \pageref*{universal_property} \linebreak
\noindent\hyperlink{the_tychonoff_theorem}{The Tychonoff theorem}\dotfill \pageref*{the_tychonoff_theorem} \linebreak
\noindent\hyperlink{relation_to_singular_cohomology}{Relation to singular (co)homology}\dotfill \pageref*{relation_to_singular_cohomology} \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}

Given two [[topological spaces]] $(X_1,\tau_1)$ and $(X_2, \tau_2)$, then the [[Cartesian product]] of their underlying sets $X_1 \times X_2$ is naturally equipped with a [[topological space|topology]] $\tau_{X_1 \times X_2}$ itself, generated from the [[base of a topology|base opens]] which are themselves Cartesian product $U_1 \times U_2 \subset X_1 \times X_2$, of [[open subsets]] of the original spaces: $U_i \subset X_i$. The resulting topological space

\begin{displaymath}
(X_1 \times X_2 , \tau_{X_1 \times X_2})
\end{displaymath}
is called the \emph{product topological space} of the two original spaces.

The [[formal duality|formally dual]] concept is that of [[disjoint union topological spaces]].

More generally, consider any index [[set]] $I$ and an $I$-indexed set $\{X_i, \tau_i\}_{i \in I}$ of [[topological spaces]]. Then the \emph{product topology} $\tau_{prod}$ or \emph{[[Tychonoff topology]]} on the [[Cartesian product]] $\underset{i \in I}{\prod} X_i$ of underlying sets is equivalently

\begin{enumerate}%
\item the topology generated from the [[sub-base of a topology|sub-base]] given by products $\underset{i \in I}{\prod} U_i$ with $U_i \subset X_i$ open, but \emph{all except one of the factors} equal to the corresponding $X_i$,

hence the topology whose open subsets are precisely those obtained as arbitrary unions of finite intersections of such subsets;


\item the topology generated from the [[base of a topology|base]] given by products $\underset{i \in I}{\prod} U_i$ with $U_i \subset X_i$ open, but \emph{all except a finite number of factors} equal to the corresponding $X_i$,

hence the topology whose open subsets are precisely those obtained as arbitrary unions of such subsets.



\end{enumerate}
This product topology is singled out by the fact that the resulting product topological space is the [[category theory|category theoretic]] [[product]] of the original space in the [[category]] [[Top]] of topological spaces:

\begin{displaymath}
\left(
     \underset{i \in I}{\prod} X_i
     ,\;
     \tau_{prod}
  \right)
  \;\simeq\;
  \underset{i \in I}{\prod} (X_i, \tau_i)
  \,.
\end{displaymath}
This means that the product topology enjoys the [[universal property]] that for any topological space $(Y,\tau_Y)$ then sets of [[continuous functions]] $\{(Y, \tau_Y) \overset{\phi_i}{\to} (X_i, \tau_i)\}_{i \in I}$ into the factor spaces are in [[natural bijection]] with continuous functions $(\phi_i)_{i \in I} \colon (Y, \tau_Y) \to \left(\underset{i \in I}{\prod} X_i, \tau_{prod}\right)$ into the product topological space.

Beware that (among others) there is also the \emph{[[box topology]]} $\tau_{box}$ on the Cartesian product of underlying sets $\underset{i\in I}{\prod} X_i$, whose open subsets are the unions of those of the for $\underset{i \in I}{\prod} U_i$ with $U_i \subset X_i$ open and with \emph{no} further restriction on the factors.

For $I$ a [[finite set]], then these two topologies coincide, but for $I$ not finite then the box topology is a strictly [[fine topology|finer topology]]

\begin{displaymath}
\tau_{prod} \subset \tau_{box}
\end{displaymath}
and hence in this case it does \emph{not} enjoy the [[universal property]] of the product topology above.

[[!include universal constructions of topological spaces -- table]]

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

\begin{defn}
\label{ProductTopologicalSpace}\hypertarget{ProductTopologicalSpace}{}
Let $\{(X_i, \tau_i)\}_{i \in I}$ be a [[set]] of [[topological spaces]]. Then their \emph{product topological space} has as underlying set the [[Cartesian product]] $\underset{i \in I}{\prod} X_i$ of the underlying sets, and has as topology $\tau_{prod} \subset P(\underset{i \in I}{\prod} X_i)$ the [[coarse topology|coarsest topology]] such that all the [[projection]] maps

\begin{displaymath}
p_i 
  \;\colon\;
   \underset{i \in I}{\prod} X_i
   \longrightarrow
    X_i
\end{displaymath}
become [[continuous functions]] (calle the \emph{Tychonoff topology}).

This means equivalently that $\tau_{prod}$ is the topology generated from the [[sub-base of a topology|sub-base]]

\begin{displaymath}
\beta_{prod}
   \;\coloneqq\;
   \left\{
      p_i^{-1}(U_i) \subset \underset{i \in I}{\prod} X_i
      \;\vert\;
      i \in I, U_i \subset X_i \, \text{open} 
   \right\}
   \,.
\end{displaymath}
\end{defn}
Notice that

\begin{displaymath}
p_i^{-1}(U_i)
  =
  U_i 
  \times
  \left(\underset{j \in I \backslash \{i\}}{\prod} X_j\right)
\end{displaymath}
and that

\begin{displaymath}
p_i^{-1}(U_i)
  \cap
  p_j^{-1}(U_j)
  =
  U_i \times U_j
  \times
  \left(\underset{k \in I \backslash \{i,j\}}{\prod} X_k\right)
\end{displaymath}
etc.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{ProductWithDiscreteFiniteTopologicalSpace}\hypertarget{ProductWithDiscreteFiniteTopologicalSpace}{}
Let $X$ be any [[topological space]] and write $Disc(\{1,2\})$ for the [[discrete topological space]] on a set with two elements. Then there is a [[homeomorphism]]

\begin{displaymath}
X \times Disc(\{0,1\})
    \;\simeq\;
  X \sqcup X
\end{displaymath}
between the product space of $X$ with $Disc(\{1,2\})$ and the [[disjoint union space]] of $X$ with itself.

\end{example}
\begin{proof}
By definition of [[disjoint union]] there is a [[bijection]] of underlying sets $X \sqcup X \simeq X \times \{1,2\}$.

By unwinding the definitions

\begin{enumerate}%
\item the open subsets of $X \times Disc(\{0,1\})$ are unions of those of the form $U \times S$, where $U \subset X$ is an open subset and $S \subset \{1,2\}$ is any subset


\item the open subsets of $X \sqcup X$ are of the form $U \sqcup V$ with $U,V \subset X$ open.



\end{enumerate}
Under the above bijection the we have

\begin{displaymath}
U \sqcup V = \left(U \times \{1\}\right) \cup \left( V \times \{2\}\right)
  \,.
\end{displaymath}
Conversely, every union of elements of the form $U' \times \{1\}$, $V' \times \{2\}$ and $W \times \{1,2\} = W \times \{1\} \cup W \times \{2\}$ is of the form $U \times \{1\} \cup V \times \{2\}$.

This shows that the above bijection of underlying sets induces a bijection of the corresponding open subsets, hence is a homeomorphism.

\end{proof}
\begin{example}
\label{}\hypertarget{}{}
For $n \in \mathbb{N}$ consider the [[Cartesian space]] $\mathbb{R}^n$ with the [[metric topology]] induced from its [[Euclidean space|Euclidean metric]] structure. Then the product topological spaces satisfy

\begin{displaymath}
\mathbb{R}^{n_1}\times \mathbb{R}^{n_2}
  \simeq
  \mathbb{R}^{n_1 + n_2}
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{InfiniteProductOfDiscreteSpaces}\hypertarget{InfiniteProductOfDiscreteSpaces}{}
Let $Disc(S)$ be a [[discrete topological space]] on a set with at least two elements. Then the infinite product space

\begin{displaymath}
\underset{n \in \mathbb{Z}}{\prod} Disc(S)
\end{displaymath}
is itself \emph{not} a discrete space.

\end{example}
\begin{proof}
The open subsets of a discrete space include \emph{all} the subsets of the underlying set. Hence we need to see that there are subsets of the [[Cartesian product]] set $\underset{n \in \mathbb{Z}}{\prod} Disc(S)$ which are not open subsets in the Tychonoff topology.

But by definition the open subsets in the Tychnoff topology are unions of products of open subsets of the factor spaces such that only a finite number of the factors is not the total space.

Since $S$ is assumed to contain at least two elements let $1 \in S$ be one of these. Then $\{1\} \subset S$ is a proper subset. Accordingly the product subset

\begin{displaymath}
\underset{n \in \mathbb{B}}{\prod} \{1\}
  \;\subset\;
  \underset{n\in \mathbb{N}}{\prod} Disc(S)
\end{displaymath}
is not open. For if it were, it would have to be the union of product subsets that contain the total set $S$ in at least one entry, which by construction it is not.

\end{proof}
\begin{example}
\label{CantorSpace}\hypertarget{CantorSpace}{}
\textbf{([[Cantor space]])}

Write $Disc(\{0,1\})$ for the the [[discrete topological space]] with two points. Write $\underset{n \in \mathbb{N}}{\prod} Disc(\{0,2\})$ for the [[product topological space]] of a [[countable set]] of copies of this discrete space with itself (i.e. the corresponding [[Cartesian product]] of sets $\underset{n \in \mathbb{N}}{\prod} \{0,1\}$ equipped with the Tychonoff topology induced from the [[discrete topology]] of $\{0,1\}$).

Then consider the [[function]]

\begin{displaymath}
\itexarray{
    \underset{n \in \mathbb{N}}{\prod}
      &\overset{\kappa}{\longrightarrow}&
    [0,1]
    \\
    (a_i)_{i \in \mathbb{N}}
    &\overset{\phantom{AAAA}}{\mapsto}&
    \underoverset{i = 0}{\infty}{\sum} \frac{2 a_i}{ 3^n}
  }
\end{displaymath}
which sends an element in the product space, hence a [[sequence]] of binary digits, to the value of the [[power series]] as shown on the right.

One checks that this is a [[continuous function]] (from the [[product topology]] to the [[Euclidean space|Euclidean]] [[metric topology]] on the [[closed interval]]). Moreover with its [[image]] $\kappa\left( \underset{n \in \mathbb{N}}{\prod} \{0,1\}\right) \subset [0,1]$ equipped with its [[subspace topology]], then this is a [[homeomorphism]] onto its image:

\begin{displaymath}
\underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\})
    \overset{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow}
  \kappa\left(
    \underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\})
  \right)
    \overset{\phantom{AAAA}}{\hookrightarrow}
  [0,1]
  \,.
\end{displaymath}
This image is the \emph{[[Cantor space]]} as a subspace of the closed interval.

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

\hypertarget{basic_properties}{}\subsubsection*{{Basic properties}}\label{basic_properties}

\begin{prop}
\label{ProjectionsAreOpenMaps}\hypertarget{ProjectionsAreOpenMaps}{}
\textbf{([[projections]] are [[open maps]])}

For $\{X_i\}_{i \in I}$ a set of topological spaces, the for each $j \in I$ the [[projection]] out of their product space into the $j$th component

\begin{displaymath}
p_j \;\colon\; \underset{i \in I}{\prod} X_i \longrightarrow X_j
\end{displaymath}
is an [[open map]].

\end{prop}
\begin{proof}
Since images preserve unions (\href{interactions+of+images+and+pre-images+with+unions+and+intersections#PreservationOfUnionsAndIntersectionsOfSets}{this prop.}) it is sufficient to check that the image under $p_j$ of a [[base fo the topology|base open]] is still open. But base opens in the product topology by definition are, in particular, products of open subsets.

\end{proof}
\hypertarget{universal_property}{}\subsubsection*{{Universal property}}\label{universal_property}

The product topological space construction from def. \ref{ProductTopologicalSpace} is the [[limit]] in [[Top]] over the [[discrete category|discrete]] [[diagram]] consisting of the factor spaces, hence the [[category theory|category theoretic]] [[product]].

For \textbf{proof} see at \href{Top#UniversalConstructions}{Top -- Universal constructions}.

\hypertarget{the_tychonoff_theorem}{}\subsubsection*{{The Tychonoff theorem}}\label{the_tychonoff_theorem}

The [[Tychonoff theorem]] states that the product space of any set of [[compact topological spaces]] (with its Tychonoff topology) is itself compact.

\hypertarget{relation_to_singular_cohomology}{}\subsubsection*{{Relation to singular (co)homology}}\label{relation_to_singular_cohomology}

The [[singular homology]] of product topological spaces is informed by

\begin{itemize}%
\item [[Eilenberg-Zilber theorem]]


\item [[Künneth theorem]]



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

\begin{itemize}%
\item [[tensor product of chain complexes]]

\end{itemize}
[[!include universal constructions of topological spaces -- table]]

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

The \emph{Tychonoff topology} is named after [[A. N. Tychonoff]].

\begin{itemize}%
\item [[Terence Tao]], \emph{Notes on products of topological spaces} (\href{http://www.math.ucla.edu/~tao/resource/general/121.1.00s/product.pdf}{pdf})


\item [[Florian Herzig]], \emph{Product topology} (\href{http://www.math.toronto.edu/~herzig/MAT327-lecturenotes08.pdf}{pdf})



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

[[!redirects product space]] [[!redirects product spaces]]

[[!redirects Tychonoff product]] [[!redirects Tychonoff products]] [[!redirects Tychonov product]] [[!redirects Tychonov products]] [[!redirects Tykhonoff product]] [[!redirects Tykhonoff products]] [[!redirects Tykhonov product]] [[!redirects Tykhonov products]] [[!redirects Tichonoff product]] [[!redirects Tichonoff products]] [[!redirects Tichonov product]] [[!redirects Tichonov products]] [[!redirects Tikhonoff product]] [[!redirects Tikhonoff products]] [[!redirects Tikhonov product]] [[!redirects Tikhonov products]] [[!redirects Тиьонов product]] [[!redirects Тиьонов products]]

[[!redirects Tychonoff topology]] [[!redirects Tychonoff topologies]] [[!redirects Tychonov topology]] [[!redirects Tychonov topologies]] [[!redirects Tykhonoff topology]] [[!redirects Tykhonoff topologies]] [[!redirects Tykhonov topology]] [[!redirects Tykhonov topologies]] [[!redirects Tichonoff topology]] [[!redirects Tichonoff topologies]] [[!redirects Tichonov topology]] [[!redirects Tichonov topologies]] [[!redirects Tikhonoff topology]] [[!redirects Tikhonoff topologies]] [[!redirects Tikhonov topology]] [[!redirects Tikhonov topologies]] [[!redirects Тиьонов topology]] [[!redirects Тиьонов topologies]]

[[!redirects topological product]] [[!redirects topological products]]

[[!redirects product topology]] [[!redirects product topologies]]



\end{document}