Consider the sequence , i.e. . I claim that the limit of this sequence is . Meaning we need to show for sufficiently large . Meaning, be made sufficiently small. This integral is which can be made smaller than any . We have shown this sequence converges to in the first topology.
Consider the same sequence in the second topology. I claim the limit is . We need to show when is sufficiently large. But this means . But the maximum value is because on . This (0) is certainly less than any epsilon thus thus it converges to 1 in the second topology.
Thue, the two fields are not equivalent.