Learn Turkish: Havuza veya hamama girip girmediğini kimse görmedi. - No one saw whether or not he entered the pool or the Turkish bath.

This is the core structure for indirect yes/no questions! It starts with the positive verb stem plus -(y)ip (girip), followed immediately by the negative verb stem plus the object participle -dik (girmediğ-). We then add a possessive suffix to show who did or didn't do the action (here, -i for "his/her"), and finally the accusative case -ni because this whole phrase is the direct object of the main verb görmedi (didn't see).

Why do havuza and hamama both end in the dative suffix -a?

Because the verb girmek (to enter) always requires the dative case (-(y)e / -(y)a) to show the place being entered. Even though the verb is locked up in the -ip -mediği construction here, it still governs the nouns before it just like a regular verb would. Therefore, both havuz (pool) and hamam (bath) take the dative -a.

The English translation says "no one saw", but the Turkish verb görmedi is negative. Why?

In Turkish, indefinite pronouns like kimse (nobody/anybody) or hiçbiri (none of them) must be paired with a negative verb when used in a negative sense. So, kimse görmedi literally means "anybody didn't see", which translates naturally to "no one saw" in English.