This is perhaps a misunderstood definition rather than a false belief, but:
"A subnet of a net $( x_\alpha )_{\alpha \in A}$ takes the form
$( x_\alpha )_{\alpha \in B}$ for some subset $B$ of $A$."
In truth, subnets are allowed to contain repetitions, and can be indexed by sets much larger than the original net. (In particular, there are subnets of sequences that are not subsequences.)
This false belief, incidentally, reinforces the false belief noted in a different answer, namely that compactness implies sequential compactness.
