Could anyone tell me what is the correct or better name for this gate/door?
They are usually used by old shops in Asian countries (possibly from 1940s to 1960s).
The photo is on this web-page.
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The Japanese variation of these are known as a folding shoji screen.
In Japanese, a 'shoji' is a wooden sliding door with translucent paper on, so a 'folding shoji' is a variation of that. I see your photo doesn't have the paper on it though.
In British English, a similar piece of furniture is known as a room divider. I see that on various online stores they advertise "Asian/Oriental Screens & Room Dividers", so perhaps that is the default descriptive name used, as 'shoji' is probably not widely known.