Is It Lawful For Police To Enter Your Home For The Door Being Open At Night?

Location: Kentucky

My mom loves to leave the door to our house wide-open all night. I tell her all the time it could get us robbed or something but I guess she values the cats outside being able to come in and out more, I dunno.

Anyway, last night she was sleeping and I had headphones on. I thought I heard someone yelling at the entrance door, but I thought it was the TV in the other room. I imagine this was probably the police asking if anyone was home.

A few seconds later, I heard voices more clearly so I took off my headphones and heard voices in the stairs saying "D***** Police Department, keep your hands up." The entrance to my house immediately leads to stair where you have to turn a corner at the top to get to the rest of the place.

I was a bit dazed so I asked who it was again. I was a bit scared it may have not been the police at first but I had nothing to defend myself so I just said okay and yelled at my mom to come with her hands up.

I said we had them up, and the turned the corner with their guns aimed down. My mom got pissy and asked what they were doing here, they responded they believed there may have been a burglary going on since the door was open at 3 AM. They asked for our IDs, but I couldn't find mine so mom gave hers and they left a bit later.

My mother wants to sue them. From the bits I looked up this seems like it's lawful, but her point was if we we're sleeping and we just had the door open because we felt like it, it could have gone bad if we were suddenly woken up by people in her house.

I dunno, I've been telling her she's gonna spend a lot of money and have a hard time on a case like this, but she seems to think its concrete so I thought I might as well ask here. What do you guys think about the encounter?