"Я у басейні."

Translation: I am at the pool.

[Bit2]

What's the difference in usage of "у" and "в" in this case

[ali.kinsella]

The biggest difference is that it's easier to say "у" before the б-sound. You can liken this to a/an in English, though it's more flexible in Ukrainian. There are specific rules on when to use which sound, but Ukrainians don't follow them in everyday speech.

[canongigue]

I heard only 'basenj', couldn't hear the last syllable 'ni'. (reported)

[greg.mort]

It's there, just very very quiet

[TobyBartels]

Any relation between ‘басейні’ and the French ‘piscine’?

[YuriMykolayevych]

They translate to one another and are distant cognates. A closer cognate is the English "basin" or French "bassin".

[ali.kinsella]

Yep.

[daguipa]

Well, with "piscine" obviously not, but with "bassin", yes.