Very smart Boycat!
This is an appropriate time to mention my cat door-opening remedy/mechanism.
You know how it is, you just get all settled and comfy on the sofa, the fire is on, the room is warming up nicely... and the cat comes in.
It leaves the door open and howling gales start coming in. You have to get up and shut the door.
You settle down again, and the cat has gone out - leaving the door open... again.
Repeat ad infinitum 'til it's time to go to bed.
I hammered a tack into the top of the door, (on the side which moves into the room) and another one on the picture rail at the top of the door frame on the same side.
Then I got a decent sized rock and tied it up in a bit of velvet which matches the curtains.
Using fishing line (which is strong and nearly invisible), I tied one end to the tack in the top of the door, then, with the rock tied to the other end, I draped the line over the tack on the picture rail.
When the door is opened, the rock rises, and after cat has come in, the weight of the rock pulls the door shut again.
You have to get used to a flying velvet thingy going up and down beside the door, but it's not too hard.