I think what you have there is a microfibre eyelet Dri-Fit garment.
This weave is often referred to as "eyelet" (or an eyelet fabric), as well as (mock) eyelet. I have not been able to find a distinction between the two yet.
It's similar to Nike's Dri-Fit fabric, and you can find it labelled Dri-Fit, as well.
Dri-Fit, like 'eyelet', refers to a weave, rather than a material, and it is apparently available in microfibre, (regular) polyester, pique, and heather. It does not only refer to the eyelet weave; 'mesh' and 'interlock' alternatives are available as well.
You can also find it named cool fit (again, if there is a distinction, it is not clear), and similar microfibre fabrics have that same "eyelet" distinction (e.g. image below).*
There might be an issue with using the term Dri-Fit, as it is possibly trademarked.