Pullback of family I know the notion of pullback of two topological spaces $X, Y$ over $B$, and wanted to know if there was a pullback of an arbitrary family of topological spaces $\{X_i\}$ over $B$.
I thought it could be the subspace of the product space such that $f_i(x_i) = f_j(x_j)$ for all $i,j$.
But then I found a definition in Algebraic topology of Bourbaki, and it uses the product of $B$ times the product of the family.  I don't see why?
Sorry for the bad formed question, I'm typing from the phone. I can clarify my doubts and use latex tomorrow if needed.
Thanks.
 A: First, your definition is correct for non-empty $I$.  Let's spell out what it says formally.  You're definition states that the pullback of $\{f_i : X_i \to B\mid i\in I\}$ is:
$$\{x\in \prod_{i\in I}X_i\mid\forall i,j\in I.f_i(x_i)=f_j(x_j)\}$$
Guessing at what definition of Bourbaki's you are referencing, it's probably something like:
$$\{(b,x)\in B\times\prod_{i\in I}X_i\mid\forall i\in I.f_i(x_i) = b\}$$
We can prove these are equivalent assuming $I$ is non-empty:
$$\begin{align}
&\{x\in \prod_{i\in I}X_i\mid\forall i,j\in I.f_i(x_i)=f_j(x_j)\} \\
\cong\ & \{x\in \prod_{i\in I}X_i\mid\forall j\in I.\exists b\in B.\forall i\in I. f_i(x_i) = b\land f_j(x_j)=b\} \\
\cong\ & \{x\in \prod_{i\in I}X_i\mid\forall j\in I.\exists b\in B.(\forall i \in I.f_i(x_i) = b)\land f_j(x_j)=b\}\\
\cong\ & \{x\in \prod_{i\in I}X_i\mid\forall j\in I.\exists b\in B.\forall i \in I.f_i(x_i) = b\}\\
\cong\ & \{x\in \prod_{i\in I}X_i\mid\exists b\in B.\forall i\in I.f_i(x_i)=b\} \\
\cong\ & \{(b,x)\in B\times\prod_{i\in I}X_i\mid\forall i\in I.f_i(x_i)=b\}
\end{align}$$
So, we see that they are equivalent... except when $I$ is empty.  For empty $I$, the former definition produces as the pullback $\mathbf{1}$ (the singleton space), while the latter produces $B$.  $B$ is arguably the more appropriate answer.  The pullback of $n$ arrows is a limit of a diagram consisting of $n$ arrows and $n+1$ objects.  The $b$ component corresponds to that $+1$ object. Consistently, the limit of a diagram consisting of a single object and no (non-identity) arrows is that object.
Even in the non-empty $I$ case, the latter definition is a little easier to work with (particularly from a constructive perspective).
