I just wanted to reply to Nailit's question, eleanor.
Nails, years ago you would have been quite right. Oil-based paint, (usually gloss) would have been preferred over water-based or matt and satin finishes.
Paint technology has leapt forward in recent years. That preference no longer applies.
It really is all about surface preparation. Not just rubbing down, but more importantly, de-greasing. Water vapour (steam) in a bathroom is loaded with body oils and other greasy yuk.
Shedders mentioned a very old traditional product - sugar soap.
It's cheap and easily scrubbed on.
It's still a very effective de-greaser.