A bedroom does not require a closet thats a old wives tale that has been propagated for about 50 years. Prior to WW-11 and Levitt Town style Tract homes most people used what was called Amouires which was a piece of . furniture and you took it with you- Built in closets were for the rich and wealthy. Toady what constitutes a bedroom is normally it has to have a window in which to escape out of in the event of a fire or natural disaster. when someone removes a wall and opens up a bedroom then its no longer considered as a functional bedroom and re-labeled to something else but if its inexpensive to re-convert back to a bedroom ( Put Wall Back Up ) then it may be called a bedroom with a cost to cure ( Typical Cost $2,000 +/- anyway most old mentors trained there FHA appraisers that a bedroom had to have a closet - funny because in my 35 plus years there is nothing in FHA handbook or Fannie or Freddie BUT the issue is market acceptance if a particular area mostly has built in closet's then a buyer may expect them BUT it's cheap to install closets and they can even be purchased already built.
The photo is of Armoire- in some very high end custom homes I have seen 25 foot long ones and ones that cost over $10,000 a cheap ones is typically $200 to $300 bucks. When asked this question by my trainees I used to tell them a Bedroom with out closets is just that and a bedroom does is designed to sleep in not hang clothes and to not get hung up on it .