Can I leave my T/C plugged in to A/C when not in use?

I currently have my T/C unloaded and sitting in my yard with a full cover on it. I have it plugged in to a/c.
If I unplug it, the carbon monoxide detector will run the battery down in about 10 days to two weeks. Then the detector starts beeping low power supply.

So I have just left it plugged in.

Is this okay? I have to disconnect and pull the battery completely out to check the water level but I have been doing this.

I am assuming that the onboard battery charger just charges when needed and keeps the battery fully charged.
Am I right in assuming this?

Just want to make sure I'm doing the right thing.

Thanks,
Zulu