Learn Hardcore Russian: Второй этаж библиотеки открыт для всех посетителей. - The second floor of the library is open to all visitors.

Второй этаж библиотеки functions as the subject of the sentence (“the second floor of the library”), and subjects in Russian take the nominative case.

What part of speech is открыт, and why is it used instead of открытый?

Открыт is a short-form adjective (also called a predicative adjective) derived from the full adjective/participle открытый. Short-form adjectives are used predicatively (to state a condition) without a linking verb, whereas the full form открытый is used attributively (to directly modify a noun).

Why is there no verb “to be” (e.g. “is/are”) in this sentence?

In Russian, the present-tense copula быть (“to be”) is normally omitted. The state expressed by открыт implies “is,” so you don’t need an explicit “is” in the sentence.

Why does открыт agree in gender and number with этаж?

Short-form adjectives agree in gender and number with their subject. Этаж is a masculine singular noun, so the short-form adjective is открыт (masculine singular). If the subject were feminine or plural, it would be открыта or открыты.

Why is для used with the genitive case in для всех посетителей?

The preposition для always governs the genitive case in Russian, indicating “for” in the sense of intended audience or purpose.

Why are both всех and посетителей in the genitive plural?

Because для requires genitive, все (“all”) becomes всех in genitive plural, and посетитель becomes посетителей. Both words agree in case and number after для.

Why is библиотеки in the genitive singular following этаж?

Russian expresses “X of Y” by putting Y in the genitive after X. Here библиотеки is the genitive singular of библиотека, so этаж библиотеки means “the library’s floor.”