There are times that we need only half onion for our cooking, especially when the onion is too big. Another case is when you are so thrifty, you only use half onion in every meal you cook.
The latter is not a good cooking idea, it gonna end up in not so delectable menu.
In case you really need only half of the onion, cut it cross wise leaving the part with roots for later use. The lower part is more mature and it will last longer than upper.
Are you convince ?
If you are observant to those rotting onions, you will notice that rotting often start from the top. Also try cutting an onion as I described. Leave it at room temperature and observe for a couple of days. Tell me if your observation contradict this article.