Note that the prerequisite is weapon focus with the weapon.
So you take weapon focus: longsword, improved crit: longsword.
If you are a dual wielder you can also take improved crit: shortsword.
But you cannot keep taking improved crit: longsword to bring your crit range down to 2-20.
He is not saying weapon focus is a prerequisite. He is using weapon focus as an example because with both weapon focus and improved critical you have to choose a specific weapon such as a longsword, dagger, and so on.
Let's look at the full text:
Attacks made with your chosen weapon are quite deadly.
Prerequisite: Proficient with weapon, base attack bonus +8.
Benefit: When using the weapon you selected, your threat range is doubled.
Special: You can gain Improved Critical multiple times. The effects do not stack. Each time you take the feat, it applies to a new type of weapon.
This effect doesn't stack with any other effect that expands the threat range of a weapon.
Note the phrases I've put in italics. The full text makes clear that you choose a single weapon, and you get to double the threat range.
A further clarification: it doesn't require one particular specific weapon, e.g. the longsword your character's father carried to war with a copper-wrapped hilt that he used to kill three dozen orcs over the course of a year. One longsword is as good as another. However, no matter how similar a bastard sword and a longsword are, they're different enough that you can't apply the feat to both.