Let $X$ be a scheme over a field $k$. For an extension field $K$ of $k$, we can change the base to obtain the scheme $X_K$. Supposedly it is possible that $X$ is irreducible while $X_K$ is reducible, indeed this leads to the notion of geometric irreducibility.
I am having trouble understanding this from a topological point of view. The way we define the base change is as the fiber product with respect to the maps $X\to Spec(k)$ and $Spec(K)\to Spec(k)$, but in particular this object $X_K$ is a fiber product in the category of topological spaces. As $Spec(K)$ and $Spec(k)$ are one point spaces, the pullback ought to simply be $X$. How can it be that the pullback $X_K$ is not homeomorphic to $X$?
I realize it is probably a very simple error I am making here.
I was making an error, from which I learn something important. Fiber products in the category of Schemes are not fiber products in the category of Topological Spaces. Since there are more objects and more morphisms in $\mathfrak{Top}$, a fiber product in that category has to be universal for more stuff than a fiber product in the category $\mathfrak{Sch}$, so even though the diagram looks the same, the universality is what is changing things here.