I am currently learning point-set topology via the book “Topology, 2nd edition” written by James Munkres. Theorem 34.3 at page 218 states that a space $X$ is completely regular if and only if it is homeomorphic to some arbitrary product of interval $[0,1]$ in the uniform topology.
First things first, the definition of Munkres of completely regular space, is $T_{3 \, \frac{1}{2}}$ (Tychonoff space).
If this theorem is true, it implies that a Tychonoff space is metrizable and that a $T_4$ space is metrizable since it is homeomorphic to some arbitrary product of interval $[0,1]$ equipped with uniform topology which is metrizable. However we can find examples of $T_4$ space that are not metrizable.
What is the issue?
What did I miss?
Best Answer
Munkres means this product $[0,1]^J$ to have the product topology, not the uniform metric topology. The product topology being the minimal one to make all projection maps continuous.
In the proof of Thm 34.1 the uniform metric is used, but that's for the countable base case (Urysohn's metrisation theorem), but it switches to the product topology later (step 2), and the generalisation 34.2 which has no separate proof is supposed to use the product topology, though Munkres is (IMO) quite unclear on this. I can understand your confusion.