Is it possible to have a fabric that just allows UV light through ?

Yes. That material is called Transol. Here is a non-technical article about it.
Here is where you can buy it.

amazing @LuckyGuy I nevah woulda thunk it.

Cool – did not know such things existed. I do think I remember seeing a pair of glasses once that allowed a person to see thru certain clothing due to something related to UV light – almost bought them but…

You can buy trendy bathing suits that will put a small suntan ‘tattoo’ on parts of your body. Typically it’s a heart shape. The result is less permanent than ink.
Not that I would know anything about this.

I know that there are special fabrics made so that you can get an “all-over tan”. I also know that if you are working on your roof a T-shirt is no protection. I go a pretty bad sunburn even though I was wearing a t-shirt and ball cap. Blue jeans apparently are good protection because I was unharmed below the waist.

I don’t know too much about the exotic fabric that Luckyguy is touting, but as a long-time sewing enthusiast and conisseur of more usual fabrics, I can tell you that Ron C is correct. There are any number of commonplace fabrics that will allow enough UV light through to get a burn if you’re in the sun long enough. They are mostly loosly woven or knit cottons, linens and lightweight man-made fabrics. Woolens and interestingly silks are better UV barriers as are densly woven cottons (like denim). Ron C, next time you’re up on a roof, try a chambray shirt over your t-shirt. The combination of one knit and one woven cotton is usually enough.

@Mountain_Crafter “next time you’re up on a roof, try a chambray shirt over your t-shirt. The combination of one knit and one woven cotton is usually enough.” Good to know but the next time the roof gets done it will be a contractor up there, not me. Actually, I’ve been thinking about a metal roof. I’m getting too old for massive do-it-yourself projects.