Question: Let $\phi:X\to X$ be bijective and continuous. Does a topology $\tau$ exist such that $$f_n\overset\tau\to f\Leftrightarrow \phi\circ f_n\overset\infty\to\phi\circ f$$ where $\infty$ designates the topology induced by the uniform norm.
Attempt: For $\phi=\log$ this is the case, by this answer. But I don't understand why: defining necessary and sufficient conditions for a sequence to converge (i.e. $f_n\to f\Leftrightarrow\cdots$) does not always induce a topology $\tau$ in which convergence ($f_n\overset\tau\to f$) corresponds to the convergence defined $f_n\to f$ (see this question).