The phrase "close tabs" refers to ending or finishing tasks, activities, or responsibilities that are ongoing or pending. It can also mean shutting down or finalizing any thoughts or concerns related to a specific matter. Full definition
Be sure you are taking steps to help your teen deal with an abusive boyfriend or girlfriend while keeping close tabs on what is going on. (verywellfamily.com)
In all likelihood, your kids won't have much money in their account and won't keep close tabs on the balance. (humbledollar.com)
You could easily close this tab in your browser and visit some other resource. (resume2017.net)