I need to show that if X is an uncountable Tychonoff space, then C[X] is not metrizable. All I've been able to show so far is that that F[X], the space of all functions with pointwise topology, is homeomorphic to (R)^X (the product) which is not metrizable, but I can't seem to get much further. Thanks
Hint: If the pointwise convergence topology was metrizable, it would be first countable, so $0$ would have a countable neighbourhood base. Thus there would be a sequence of open neighbourhoods $U_i$ of $0$ such that every neighbourhood of $0$ contains some $U_i$.