Return of "last months rent"

I have been renting a room in my home for the past 2 months. Unexpectedly my renter told me he would be leaving at the end on the month. He told me this on Jan 22, giving me 9 days notice.
We have no rental contract. The only thing that was signed by both of us was a receipt for first and last months rent. No deposit was collected.
Since he already paid the rent for January at the beginning of January he feels that I should return the “last months rent” that he paid when he moved in.
What legal standing do I have not to return this money.
I was thinking of telling him that I would prorate it if I found a new roommate to move in before the end of January.
Any thoughts would be helpful.
Andy

Your LL/T laws?

Every state has landlord/tenant law and you need to Google yours and read it to see what it says about return of deposit, etc.

e.g.: WA State law says LL may retain but has to give T written notice as to why, etc and is very specific about this…and should LL do it wrong or fail to do it then he (LL) must return full deposit and pay attorneys fees and penalty should he fail to do so.

I’d go to Google and put in: “KY (or whatever state the prop is in) LandlordL-Tenant law”

It does depend on your regulations however:

If I use my regulations as an example (most will be different but similar) last month rent (LMR) is only to be used for the last month and nothing else. Damage deposit is a different matter.
Because LMR is only to be applied to the last month of occupancy every code states the necessary notice period required by a tenant to vacate. Look this up in your state code and inform your tenant of the period required. Our code as an example states a tenant must give 60 days notice and the LMR would then be applied to their actual last month as defined by the code. To use your case as a example giving 9 days notice before the end of January means your tenant (in our area) would be required to pay for February rent and the LMR deposit would be applied to March. Notice is for full month periods which means if a tenant gave notice, for example, on the second of February they would be liable for rent to the end of April.
Every area is different so check your state codes. You will find a specific notice period applies and the likely hood is your tenant has forfeited their LMR deposit.

Thank you for taking the time to answer my question. I found it very helpful and feel confident that I can now resolve the matter.
Thanks again.

You certainly do need to check your local laws, but the majority of those with which I am familiar would require the tenant to give at least one full rental period of notice.

I would go with jim’s post. And would also suggest you to read about the landlord tenant laws of your state, that would give you a clear picture of what you have to do.