Definition of $X \times_Y X$ Let $X,Y$ be topological spaces and let $f: X\to Y$. 
Questions:


*

*What is $X\times_Y X$?

*What is the map $\Delta_f: X \to X\times_Y X$?

 A: The space $X\times_Y X$ is a pullback, a special case of more general limits of topological spaces. See any textbook about category theory, for example Mac Lanes's Categories for the Working Mathematician.
Given a diagram maps $f:X\to Y$ and $g:Z\to Y$, the pullback $X\times_Y Z$ is a space which comes together with maps $k:X\times_Y Z\to X$ and $l:X\times_Y Z\to Z$ such that


*

*$fk=gl$

*For any space $W$ and maps $k':W\to X$ and $l':W\to Z$ such that $fk'=gl'$, there exists a unique map $m:W\to X\times_Y Z$ such that $km=k'$ and $lm=l'$.


It is easy to show that $X\times_Y Z=\{(x,z)\in X\times Z\mid f(x)=g(y)\}$ with the topology as a subspace of the product $X\times Z$, and the maps $k$ and $l$ are the projections. Given $k'$ and $l'$ as above, the unique map $m$ sends $w$ to $(k'(w),l'(w))$.
For an example of a pullback, let $f:X\to Y$ be any map and let $g=i:B\to Y$ be the inclusion of a subspace $B\subseteq Y$. Then $X\times_Y B$ is $\{(x,f(x))\mid f(x)\in B\}$, the part of the graph of $f$ which meets $B$, and this is actually homeomorphic to the preimage $f^{-1}(B)$.
If $f=g:X\to Y$, then $X\times_Y X$ is the subspace of $X\times X$ consisting of those pairs $(x,x')$ such that $f(x)=f(x')$. This is actually nothing but the equivalence relation determined by $f$.
Every map $m:X\to X\times_Y X$ is determined by maps $k$ and $l$ on $X$ such that $fk=fl$. So the map $X\to X\times_Y X$ is likely to refer to the map induced by the identity $\mathbf 1_X$, and this map sends $x$ to $(x,x)$.
