关 (traditional 關) was a door (门) with a bolt and threads barring it shut — the opposite of 开. It means to close, and also a mountain PASS or barrier (a 'gate' through the mountains, like 山海关, a famous Great Wall fort).
关 is the close to 开's open: 关门 (close the door / a shop closing down), 关上 (shut it, with 上), 关灯 (turn off the light). Its 'barrier' sense lives in 海关 (customs) and the historic mountain 关. But the warmest 关 is 关心 (guānxīn, 'to close around the heart') = to care about someone, and 关系 (guānxì) = relationship/connection — one of the most important social concepts in China. To 'close around' someone's heart is to care for them.
A door barred shut — the opposite of 开: to CLOSE.
关心 ('close around the heart') = to care about someone. And 关系 (guānxì), 'connections/relationships', is one of the most powerful ideas in Chinese society.