Wrong things I have believed for a long time

• I made it up in my head that the composition of

$$M_4(R)\cong M_2(M_2(R))\stackrel{\det }{\to }{M}_2(R)\stackrel{\det }{\to }R$$

is the det map of $M_4$! This is not true!

• Compactness $⟹$ sequential compactness for topological spaces however neither of these two properties imply the other.
• Cauchy integral formula (re)constructs a holomorphic function for any boundary value $f\in \mathcal{C}(S^1)$. However, in truth it does construct a holomorphic function $F$ but it may not agree with $f$ on $S^1$ at all!

$$\partial \bigcap _{n\ge 1}{A}_n\subseteq \bigcup _{n\ge 1}\partial A_n$$

Some general misconceptions, that I keep in my head for safekeeping:

• exact sequences do not split in general except in the category of vector spaces!