big list - Examples of common false beliefs in mathematics

Some undergraduate common false beliefs that I found

(1) If $H$ is a subgroup of $mathbb{Z}$ and $H$ and $mathbb{Z}$ are isomorphic, then $H = mathbb{Z}$;

(2) In a metric space every two open balls are homeomorphic;

(3) For $p in [1, infty]$, $L^p(X, mathfrak{M}, mu) = left{ f in mathbb{C}^X : int_X |f|^p , d mu < infty right}$ is a $mathbb{C}$-normed vector space, with the norm $lVert f rVert_p = (int_X |f|^p , d mu)^{1/p}$.

Belief (1) is very naive, for every nontrivial subgroup of $mathbb{Z}$ is of the form $n mathbb{Z}$, all of them isomorphic with $mathbb{Z}$. For (2) people tend to think of normed vector spaces and forgets the discrete metric spaces. For (3) some people just forget that one have to consider the quotient space, where the classes $[f]=[g]$ iff $f=g$ $mu$-almost everywhere.

Belief (1) is very naive, because every nontrivial subgroup of $mathbb{Z}$ is of the form $n mathbb{Z}$, all of them isomorphic to $mathbb{Z}$. For (2) people tend to think of normed vector spaces and they forget the discrete metric spaces. For (3) some people just forget that one have to consider the quotient space, where the classes $[f]=[g]$ iff $f=g$ $ mu$-almost everywhere.