Can I as landlord, keep a tenants security deposit if they move on the first of the month, then, inform me that they are moving out two weeks later?

They moved out, but only told you that they had moved two weeks later? Did they pay rent for that month? Did they have a lease?

You as the landlord have to follow the law as to the return of the deposit. Generally the law states that you have a period to return the deposit or a accounting of how it was used.

If a tenant owes rent the deposit can be applied towards that unpaid rent.

You haven't given us enough info to tell what they might owe.

This is in Illinois, they moved in on the 3rd of the month,then ten days later they informed me that they are moving out.

You have 30 days from when you take possession of the unit to return the deposit or an accounting of how it was used in whole or in part.

Did they sign a lease? If so for how long? They are responsible for the rent until the end of that lease. What does the lease say about early termination or notice of termination? IL law requires 30 day notice of termination. So, at the least, they will own rental until 30 days after they informed you of vacating.

If they have a lease, they will owe you rental until the lease expires or you find another tenant, whichever comes first.

So within 30 days of you taking possession of the unit, which would be when they notified you they were moving or when they actually moved, whichever is later, you give them a notice of what you used the deposit for (unpaid rent, repairs, etc.)

Also, generally, a tenant has to give the landlord some required notice that the tenant is going to move out. I don't know off the top of my head what that is in Illinois, but in many places it's 30 days. So, if they gave you notice on the 13th of August, they would owe rent until at least September 12th, or until the end of September (depending), or even later if there is a lease. So if they fail to pay this when due, you could take it out of the deposit.