My question is: **Does the forgetful functor F:(Mfd) $\to$ (Top) preserve pullbacks?**

Detailed explanation is following.

A pullback is defined as a manifold/topological space satisfying a universal property: a manifold/topological space $Z$ (more precisely a diagram $Y \overset{g}{\leftarrow}Z\overset{g'}{\rightarrow}Y'$) is said to be a pullback if for any diagram $Y \overset{h}{\leftarrow}W\overset{h'}{\rightarrow}Y'$, there is a unique smooth/continuous map $u:W \to Z$ such that $h=g \circ u$ and $h'=g' \circ u$.

Now **suppose that we have a manifold $Z$ which is a pullback** of the diagram $Y \overset{f}{\rightarrow}X\overset{f'}{\leftarrow}Y'$ in (Mfd). Then I wonder whether the manifold $Z$ is a pullback in (Top) or not.

Notes:

- (Top) is the category of topological spaces and continuous maps.
- (Mfd) is the category of smooth manifolds and smooth maps.
- It is well known that a pullback is given by a submanifold $$Y \times_X Y' = \lbrace (y,y') \in Y \times Y'\mid f(y)=f'(y') \rbrace$$ of $Y \times Y'$ if either $f$ or $f'$ is a submersion. But we do not assume that a pullback is of this form. (If a pullback is always given by this form in (Mfd), then this implies that the forgetful functor preserves pullbacks.)
**I am NOT asking about an example of a pullback in (Top) which is not a manifold.**Sorry for my confusing question.

existsin Mfd even if the topological pullback $Y\times_X Y'$ is not a manifold. $\endgroup$ – Eric Wofsey Mar 12 '13 at 13:20