It is well known that normality is very easily destroyed by taking Cartesian product. In a previous post, we showed that the Cartesian product of the Sorgenfrey Line with itself is not normal. Here, we present an example showing that the uncountable product of metric spaces can fail to be normal. Specifically we show that the product is not normal where is the set of all natural numbers and is the first uncountable ordinal. The set is the first infinite ordinal and is identical to .
For , let for each where , for at most one . It is clear that and are disjoint and closed. We show that and cannot be separated by disjoint open sets.
Let and be open sets in such that and . We will produce a function such that .
For any finite set and for any , let be defined as the following:
Because each factor in the product space is the discrete space , is a basic open set at the point .
Define an inductive process producing a sequence of finite sets , a sequence of functions such that for each , , , .
To start off, let be the zero function on . Since , there is some finite such that , . Define by for all and for all .
Now consider the step in the induction process where . Since , choose some finite such that . can be chosen in such a way that it extends . That is , . Now, define by for all and for all . To help with a later step, note that:
for all and for all .
Let . Define by for all and for all . Since , there is some finite such that . Since is finite, choose some integer such that . Now define by for all , for all , and for all .
Note that the function agrees with on the finite set . Thus . Note that the function agrees with on the finite set (see ). Thus . The proof that is not normal is now complete.