\chapter{Constructions of topological spaces}


Now that we are familiar with the notion of a topological space and what they represent, it is useful to look at some general methods that can be used to construct new topological spaces from existing ones.

\section{Topological subspaces}

To extend a topology to a subset of a topological space, we introduce topological subspaces.

\begin{definition}
A topological subspace of $(X,\mathcal{T})$ is a topological space $(Y,\mathcal{T}_{Y})$.
\begin{itemize}
\item $Y \subseteq X$ is a subset of $X$
\item $\mathcal{T}_Y = \{ Y \cap U : U \in \mathcal{T}\}$ is the induced topology for the topological subspace 
\end{itemize}
\end{definition}

Indeed, topological subspaces always form a topological space, however its topological properties aren't necessarily (and often aren't) the same as the original space. That said, much can be said regarding a topological subspaces' relationship with the original space.

\begin{proposition}
Let $Y$ be a topological subspace of $X$. A set if closed in $Y$ iff it is of the form $Y \cap F$ where $F$ is closed in $X$.
\end{proposition}


\begin{proposition}
Let $Y$ be a topological subspace of $X$. If $\mathcal{B}$ is a basis for $\mathcal{T}$, then $\mathcal{B}_Y = \{ B \cap Y : B \in \mathcal{B} \}$ is a basis for $\mathcal{T}_{Y}$
\end{proposition}


\begin{proposition}
Let $Y$ be a topological subspace of $X$. If $Y$ is open in $X$, then open sets in $Y$ are open sets in $X$.
Let $Y$ be a topological subspace of $X$. If $Y$ is closed in $X$, then closed sets in $Y$ are closed sets in $X$.
\end{proposition}


\begin{proposition}
Let $Y$ be a topological subspace of $X$ and $A \subseteq Y$/
\[\mathrm{cl}_{X}(A) \cap Y = \mathrm{cl}_{Y}(A)\]
\end{proposition}





\section{Product topological spaces}

\begin{definition}[Box topological space]
Let $(X_i , \mathcal{T}_i)_{i \in I}$ be a family of topological spaces. The \emph{box topological space} $\prod_{j \in I}X_j$ is the topological space generated by the following basis.
	\[\mathcal{B} = \{ \prod{j \in I} U_j : U_j \in \mathcal{T}_j \}\]
Essentially, the cartesian product of open sets from each $X_j$ is a basis element.
\end{definition}

finite products mean product topology equals box topology


\begin{proposition}
Let $(\prod^{n}_{i=1} X_i, \mathcal{T})$ be a box topological space, then all $\mathcal{T}$ is the coarsest topology such that all projection functions are continuous.
\end{proposition}

This result doesn't generalize for infinite box topological spaces.


\begin{definition}[Product topological space]
For each $i \in \mathbb{N} \cap[1,n]$, let $(X_i , \mathcal{T}_i)$ be topological spaces. The \emph{box topological space} $\prod_{j \in I}X_j$ is the topological space generated by the following basis.
Let $(\prod^{n}_{i=1} X_i, \mathcal{T})$ be a box topological space, then all $\mathcal{T}$ is the coarsest topology such that all projection functions are continuous.
\end{definition}


When we consider infinite products, the projection functions may not be continuous.
We define product topological spaces to be more robust.



\begin{definition}[Box topological space]
projection functions are all continuous.
\end{definition}


\section{Quotient topological spaces}



How can we formally define the idea of constructing a new topological space  'gluing' points of the original space?

The idea is to use equivalence relations as our 'glue'; we want points within the same equivalence class to be seen as the same point (i.e glued together)

Quotient spaces do exactly this; the term 'quotient' appears because by using equivalence clases, we partition (i.e 'divide') our original set.

\begin{definition}[Quotient space]
Given a topological space $(X,\mathcal{T})$ and equivalence relation $\sim$ on $X$, let $X / \sim$ be the set of equivalence classes and $q : x \to X /\sim$ be a surjective function defined as $q(x) = [x]$, called the \emph{quotient map}.

The \emph{quotient topology} $\mathcal{T}_{X / \sim}$ is the topology  $\mathcal{T}_{X / \sim} = \{ V \in X / \sim : q^{-1}(V) \in \mathcal{T}\}$. The topological space $(X / \sim , \mathcal{T}_{X / \sim})$ is the \emph{quotient space induced by $\sim$}.
\end{definition}





\section{Final topology}

We can generalize the idea of our quotient space 

\begin{definition}[Final topology]
Given a set $X$ and topological spaces $(Y_i,\mathcal{U}_i)$ with functions $f_i : Y_i \to X$, the \emph{final topology on $X$} is the finest topology $\mathcal{T}$ on $X$ such that all $f_i$ are continuous.
\end{definition}
