No, not so. When looking at homes, I have clients tell me, matter-of-factly that a bedroom must have a closet if it is to be considered a bedroom. Without a closet, they say, it is just an office or den. I hear the same thing from my fellow local Realtors.
I'm not sure when or how this became the accepted definition, but I did my own research to find out if there is a legal definition of what a bedroom must include. First, I checked our local MLS rules to see if there was any authority. The rules are silent on the definition of a bedroom and how bedrooms are listed in the MLS. So, I went to have a look at the San Diego Municipal Code. Under the section containing definitions (section 113.0103), a "Bedroom" means:
an enclosed space within a dwelling unit
that is designed or could be used for sleeping and has a
permanent door permitting complete closure and
separation from all kitchen, living room and hallway areas.
A room or other enclosed space is not considered a bedroom
if it is the sole access to another bedroom.
No closet is needed in order for a room to be a bedroom. Now this definition is only applicable within the City of San Diego (section 11.0104). In other areas, a closet may be necessary.
It actually makes sense that a closet is not required in order to be considered a bedroom. Older homes didn't always include a built-in closet. Instead, the occupants used an armoire or other type of free-standing closet. Ikea carries a bunch of different stand-alone closets/wardrobes just for this purpose.