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Verb: keep an eye on
  1. Follow with the eyes or the mind
    "Keep an eye on the baby, please!";
    - watch, observe, follow, watch over

Derived forms: keeps an eye on, keeping an eye on, kept an eye on

Type of: check, check into, check out, check over, check up on, go over, look into, sus [Brit, informal], suss [Brit, informal], suss out [Brit, informal]