I think it is Veronica persica or something related.

Looks like mouseear chickweed... see link.

Cameron

Here is a link that might be useful: weed photos in NC

won, I tried to make it mouseear chickweed, too, but it doesn't appear to be crenated like this weed. Esh, I think you have it. Maybe I'll let a couple of them bloom to be sure? Thank you both so much.
Pam

Cameron, thankyou for the photos of winter weeds. I had a huge stand of what I now know was hairy bittercress. Every year I try to get it before it goes to seed but it's darn near impossible.

Those photos are interesting. I have all of them.

I sometimes volunteer in a wonderful native plant garden, and Veronica is our second worst weed. First one? Italian arum! It is a beautiful plant, but it can really take over. If you have it, be aware of the awful potentials over a long period of time.

I have the Johnny Jump-ups everywhere! At least they are cute weeds.

I've been digging up the hairy bittercress for a month now, but I keep seeing it everywhere, too. Flowering now, no less.

Cameron

ncrescue..so THAT'S what I have ! Italian arum. It was a houseplant given to me and I tossed it on the compost pile in the woods. Didn't plant it. Now it hasn't grown bigger or spread but it's a dependable and interesting rich green against the leaves.
What I have out there in native abundance is mayapples. Every tree that falls down brings more dappled light and the mayapples spread by underground shoots. Used to try and avoid stepping on them but now there are so many I don't watch where I'm stepping anymore.

Cameron, I agree...Johnny Jump Ups are great weeds! Dottie, I swear that hairy bittercress comes out of the ground with a bloom on it! :)
Pam

Not only does it arrive with buds breaking but they must bloom overnight and pollinated by who knows what mini-bug because even the teeniest bittercress plants have a seedpod that invariably explodes as you threaten to pull it out of the ground.
Bad enough they're in all my borders but now they are growing happily in the moss that covers a side of my terrace foundation. Hmm..once I get the hairy bittercress out I wonder if I should try throwing alyssum seeds onto the moss?

Moss is the best place for seeds to germinate, I swear!

As the season for hairy bittercress winds down, I notice that the plants mature faster (as you said even the tiny ones get flowers and seed pods) making our jobs even tougher to find those before they go to seed.