Plant ID - low growing weed

I've noticed this weed that grows as a ground cover. The first photo is taking over some bare dirt near the wofati. The second is filled in, much more carpet like, on the path between the office and the house (so it can take some trampling). The last photo is of the tiny flower on this plant (sorry i couldn't get the focus better). These flowers are only 1-2mm across. I'm hoping to find out the name to learn more about it. I suspect that this may be a good plant to seed into our finished construction areas. Thanks for any help.

I don't have a name for that plant, but it's one of my favorites. It grows on gravel driveways around here, in areas that get too much traffic for grass to grow, but not enough to outright crush this plant. I love it as a ground cover!

It looks similar to a local plant i have identified as A member of the euphorbia or spurge Genus. Maybe it is a spurge weed you got there.

I call that plant doorweed. I believe it is a polygonum.

Most likely Polygonum arenastrum otherwise known as knotweed! Tiny white/green axillary flowers, papery leaf sheaths and dense prostrate growth. I actually just identified this in my area this week so I am not as familiar with it....yet. Plants for a future has this to say about knotweed.

As far as I can tell it is an incredible ground cover for compacted areas. It grows at my parents house in the high traffic dog yard and on a steep slope to boot. I think it could be a good plant to incorporate into some designs.

Thanks for all of your great plant ID posts! Just so happens that most of the species you post coincide with the species blooming in my area so you're always helping me with identification.

Thanks Akiva and Kai! I think you're right on this one. Zach, spurges have milky sap, and after checking, this one didn't. Although PFAF warns that other members of this genus have oxalic acid, this one doesn't taste lemony to me. I'll have to keep an eye on it and see if i can tell when the seeds are ripe.

That looks like the one we call Knotgrass. Um... because it's not grass, I suppose.