Solution:

We check each option one by one: 1. \(\{x \in \mathbb{N} : (x-1)(x-2)=0\} = \{1,2\}\), which is a finite set.

2. \(\{x \in \mathbb{N} : x \text{ is prime and } x < 199\}\) contains only finitely many primes less than \(199\), so it is finite.

3. \(\{x \in \mathbb{N} : x^{5}-1=0\} \;\Rightarrow\; x^{5}=1 \;\Rightarrow\; x=1\). Thus the set is \(\{1\}\), which is finite.

4. \(\{x \in \mathbb{N} : x \text{ is odd}\} = \{1,3,5,7,\dots\}\), which is an infinite set.

Answer: Option (4).