المكتبة العامة قرب محطة القطار.

Because this is a present-tense nominal sentence in Arabic. In Modern Standard Arabic, you usually do not say a separate word for is / are in sentences like this.

So Arabic can say:

المكتبة العامة قرب محطة القطار

where English needs:

The public library is near the train station.

If you wanted a past-tense version, then Arabic would use a verb, for example كانت for was.

Because in Arabic, adjectives normally come after the noun they describe.

So:

• المكتبة = the library
• العامة = public

Together:

• المكتبة العامة = the public library

This is the normal Arabic order: noun + adjective.

Because Arabic adjectives usually agree with the noun they describe in several ways, including definiteness.

Here, المكتبة is definite because it has ال. So the adjective must also be definite:

• المكتبة العامة = the public library

If the noun were indefinite, the adjective would also be indefinite:

• مكتبة عامة = a public library

So the second ال is not extra or optional here; it is part of normal adjective agreement.

Because مكتبة is a feminine singular noun, and the adjective has to match it.

Arabic adjectives usually agree with the noun in:

• gender
• number
• definiteness
• and, in fully vocalized Arabic, case

So since مكتبة is feminine singular, the adjective appears in the feminine singular form:

• عام = general / public, masculine
• عامة = general / public, feminine

That is why you get المكتبة العامة.

It works like the English word near.

In school grammar, قرب is often treated as an adverbial expression of place, and in actual use learners can think of it as a preposition-like word meaning near.

So:

• قرب محطة القطار = near the train station

In this sentence, that whole phrase tells you where the library is, so it functions as the predicate of the sentence.

Because قرب can directly be followed by the noun phrase after it.

So this is natural:

• قرب محطة القطار = near the train station

But if you use the adjective قريب / قريبة, then you usually use من:

• المكتبة العامة قريبة من محطة القطار

That also means The public library is near the train station.

So both patterns are possible:

• قرب + noun
• قريب / قريبة من + noun

Because this is an iḍāfa construction, often called a construct phrase.

The pattern is:

• noun + noun
• literally something like station of the train

So:

• محطة القطار = the train station

In an iḍāfa:

• the first noun usually does not take ال
• the second noun determines whether the whole phrase is definite

So محطة القطار is correct, while المحطة القطار is not the normal iḍāfa form.

That is a very common difference between English and Arabic.

Arabic often expresses English compound nouns using iḍāfa. So:

• محطة القطار literally = station of the train
• natural English = train station

Because القطار is definite, the whole phrase becomes definite:

• محطة القطار = the train station

If you made the second noun indefinite, the whole phrase would be indefinite too:

• محطة قطار = a train station / a railway station

So the definiteness of the second noun matters a lot.

This is an excellent thing to notice.

In pause, ة is usually pronounced like -a:

• مكتبة → maktaba
• العامة → al-ʿāmma

But in an iḍāfa construction, the ة is pronounced as t before the next word:

• محطة القطار → maḥaṭṭat al-qiṭār

So although the word is written محطة, in this phrase you hear the t sound because it is linked to the following noun in the construct phrase.

Yes. That is a very natural alternative.

Compare:

• المكتبة العامة قرب محطة القطار
• المكتبة العامة قريبة من محطة القطار

Both mean essentially the same thing: The public library is near the train station.

The first uses قرب. The second uses the adjective قريبة plus من.

The second version may feel a little more explicit to learners because it looks more like a full subject + predicate adjective structure.

A fully vowelled version would be:

المكتبةُ العامةُ قربَ محطةِ القطارِ

This shows the case endings:

• المكتبةُ — nominative
• العامةُ — nominative, agreeing with المكتبة
• قربَ — an adverbial expression of place
• محطةِ القطارِ — genitive after قرب in this construction

In everyday reading, especially in unvowelled text, learners usually do not pronounce all of these endings, but it is useful to know they are there in formal grammar.