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Examples of common false beliefs in mathematics.
Hopefully this is not overly controversial, but I thought it would be instructive to compile a list of errors which are commonly (or at least not too uncommonly) made in higher level mathematical research and published mathematical works (i.e. research papers, books, etc). If nothing else this could serve as a warning to the rest of us.
Here I would like to keep the focus on mathematical errors, oversights, or misunderstandings, and not those which are related to typographical, grammatical, or purely historical issues. To start the list and to give some idea what types of errors I am thinking of, here are a few examples:

1. Confusing the finite field $\mathbb{F}_{p^k}$ for $k>1$ with the ring $\mathbb{Z}/p^k\mathbb{Z}$.

2. Assuming that every open set in $\mathbb{Q}_p$ is also closed (true for balls but not in general).

3. Assuming that the hypothesis in a conditional statement is a necessary condition, just because a weaker hypothesis does not necessarily imply the conclusion.
One remark, since this is intended solely for our own edification, it is probably better to avoid mentioning specific references, for the obvious reason.
Also, if anyone can think of better tags for this question please go ahead.