Question

Here's my list of false beliefs:

• If $U$ is a subspace of a Banach space $V$, then $U$ is a direct summand of $V$.
• If $M/L, L/K$ are normal field extensions, then the same is true for $M/K$.
• Submodules/groups/algebras of finitely generated modules/groups/algebras are finitely generated.
• The Krull dimension of a subring is at most the Krull dimension of the ring.
• The Krull dimension of a noetherian domain is finite.[also surprising for Leibniz prize winners]
• If $A \otimes B = 0$, then either $A=0$ or $B=0$.
• If $f$ is a smooth function with $df=0$, then $f$ is constant.
• If $X,Y$ are sets such that $P(X), P(Y)$ are equipotent, then $X,Y$ are equipotent.
• Every short exact sequence of the form $0 \to A \to A \oplus B \to B \to 0$ splits.
• $R[[x,y]] = R[[x]][[y]]$ as topological rings.
• $R[x]^* = R^*$, even if $R$ is not a domain.
• Every presheaf on a site has an associated sheaf. (Hint: the index category of the usual colimit has to be essentially small!)
• (Co)limits may be computed in full subcategories. For example, $Spec(\prod_i R_i) = \coprod_i Spec(R_i)$ as schemes because $Spec$ is an antiequivalence.
• Every finite CW-complex is compact, thus every CW-complex is locally compact.
• The smash product of pointed spaces is associative (this is even false for CW complexes when you don't use the compactly-generated product!), products commute with quotients, and so on: Topologists assume that everything behaves well, but sometimes it does not.