\part{Constructions of topological spaces}

\chapter{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 for a topological space, however its topological properties may differ from the topological 'superspace', depending on the properties of the set the subspace is formed from.

That said, there are many relations topological subspaces have with its superspace.

\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}





\chapter{Finite product topological spaces}

\begin{definition}
	For each $i \in \mathbb{N} \cap[1,n]$, let $(X_i , \mathcal{T}_i)$ be topological spaces. The \emph{box topological space} $\bigtimes_{j \in I}X_j$ is the topological space generated by the following basis.
	\[\mathcal{B} = \{ \bigtimes_{j \in I} U_j : U_j \in \mathcal{T}_j \}\]
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



\chapter{Quotient topological spaces}




\chapter{Final topologies}


