I have a question about product topology.
Suppose $I=[0,1]$, i.e. a closed interval with usual topology. We can construct a product space $X=I^I$, i.e. uncountable Cartesian product of closed interval. Is $X$ first countable?
I have read Counterexamples of Topology, on item 105, it is dealing with $I^I$. I do not quite understand the proof given on the book. Can someone give a more detail proof?