To "keep an eye on" someone is to watch that person carefully. Example: "Would you keep an eye on my son for me while I am gone?" Reply: "Sure." You take a share of responsibility for something when you agree to keep an eye on it. Example: "I have to run to the bathroom. Can you keep an eye on my suitcase while I am away?" Reply: "No problem." "Keeping an eye on" someone can also mean watching to make sure that person does not make a mistake. Example: "Be careful what you do today. I'll be keeping an eye on you."