I think, there are different types of false beliefs. The first kind are statements which are quite natural to believe, but a moment of thought shows the contradiction. Of this type is the sin-example in the opening post or a favorite of mine (also occured to me):
- The underlying additive group of the field with $p^n$ elements is $\mathbb{Z}/p^n\mathbb{Z}$.
The other type is also quite natural to believe, but one has really to think to construct a counter example:
- Every contractible manifold is homeomorphic to $\mathbb{R}^n$.
- Every manifold is homotopy equivalent to a compact one.
- Quotients commute with products in topological spaces.
- Every connected component of a topological space is open and closed. Or related to this:
- To give a continuous action of a topological group $G$ on a discrete space $X$ is the same as to give an action of the group of connected components of $G$ on $X$.
