My yard has been plagued by this tiny viney weed. The leaves are in sets of two set about a 1/2 inch apart. The stem is very fine and delicate and ends in tendrils. The root is very fine and shallow and easily pulled. There are multiple stems and each of those also branches off so that the plant becomes a mass that smothers other plants. The blooms are very tiny, probably less than a quarter inch. They are white and resemble a pea blossom. I am attaching several picture that I took. Thank you for any help with identifying this plant.
Andrey, thank you very much. That is exactly the weed I have growing all over everything in my yard. Thank you again.