  We learn mathematics subjectively and must apply it objectively. But
sometimes, we apply it subjectively by using wrong intuitions which may be
elusive to our eyes. The aim of this note is to disclose the secretes of two
kinds of these false intuitions and the opportunities they may provide. We
first discuss the wrong assumption which says that each topology is uniquely
determined by studying a very bad phenomenon happening in dealing with weak*
topology. Then, we consider the problem of completing a normed space under two
comparable norms, one being smaller than the other. Here, we show that the
common belief contending that smaller norms give rise to larger completions is
wrong. We then pose some serious questions arising from these wrong intuitions.
As we will see finding fallacies are as important as major mathematical
activities like proving and disproving.
