I noticed there is an extra T pipe under my vanity for cold water. It goes to basement and it looks like there is a hole in it as well. I cannot understand what does it do? It looks like there is constant water that goes inside it (I can see it from the small hole it has).
The strange T part in close look:
It's a trap primer. It allows just a little bit of water to be bypassed from the usual horizontal path (headed to your sink) and sends it out the vertical pipe, where it ends up keeping a trap (often in a floor drain) from drying out and allowing sewer gas into your house.
Here's a photo of one from a vendor. https://www.sustainablesupply.com/products/zurn-trap-primer-bronze-pipe-dia-1-2-in-z1022-xl-1-2-ip-c1912466
And another one (from https://www.wilmar.com/Sku/15-1843/mifab-continuous-flow-trap-seal-primer-with-12-in-connections-779897963421-mi-tsp-1) that looks even more like the one in your photo:
(Thanks to Greg Hill and Ecnerwal for pointing me in the right direction with their comments.)
