Would something like I've linked below work?

Here is a link that might be useful: Latch

No ... that's plastic and it would be hammered to death by the persistent one.
We need metal.

It's also not self-latching. You have to remember to flip the wings up.

In the description, it says that it automatically locks. I think the wings pop back up automatically. Couldn't you mount it high enough so that it cannot be reached by the persistent one?

Never mind...in reading the reviews on Amazon, it appears that the adhesive on these is rather weak. Owners suggested screwing the the device into the wood which wouldn't be too good for your closets.

This post was edited by Fun2BHere on Sun, Jun 22, 14 at 22:25

An auto latching screen door lock may work. You need some sort of lever to open the door, though.

Here is a link that might be useful: Mortise Lock

Actually I want to see a picture of the persistent cat. :)

If the doors NEVER open for them, they'll be less persistent.

Here's one ...