It's worth having a polite, businesslike discussion about how you can work together to save each other some time and money.
It varies from state to state. Normally you have to give 30 days notice. If you don't, you have to pay for the next month. In your case, if you didn't want to be reponsible for January's rent, you needed to let them know before November 30th. Then you pay December's rent, and be out by December 31st. If you're not out until January 10th, they can't rent it to someone for the whole month of January. All of this should be in your lease, read it carefully.
In your situation, you are being charged for all of January (a full month's rent), so you don't have to be gone by then. I would tell her that I will be gone by the 10th, so I want to pay less rent. She probably won't go for it, but there's nothing you can do. But that means you don't have to be out until January 31st. Again, read your lease.
Yes, you owe the rent for the entire month of January since that is when your lease ends.
You agreed to pay rent for your entire lease term. Leaving earlier or giving notice does not relieve you from your obligations to pay as agreed. Your landlord is correct.
Yes, of course they can. You have a lease saying you will pay through 1/31, so you do. Period.
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I am moving into a new apartment on the 2nd of January. The lease at my current complex isn't up until the 31st of January. My current office has been given legal notice, and knows I will have everything moved out and keys turned in before the 10th of the month. My question is, can she legally charge me an entire month of rent if I will not be occupying this space the entire month This is my 1st ever apartment so I'm not quite sure how to handle things without being cheated.