Learn Turkish: Şemsiyen açık mı? - Is your umbrella open?

The base word for umbrella is şemsiye, which ends in a vowel. When adding the 'your' suffix, which is -(i)n, the connecting vowel is dropped to avoid having two vowels next to each other. So, we just add -n to get şemsiyen.

Do I need to use the word 'senin' (your) before 'şemsiyen'?

No, you don't have to! The -n at the end of şemsiyen already tells us it is 'your' umbrella. You could say Senin şemsiyen açık mı? if you really wanted to emphasize 'your' umbrella compared to someone else's, but it is perfectly natural to drop the pronoun.

Where is the word for 'is' in this sentence?

There isn't one! In Turkish, there is no direct equivalent to the present tense verb 'to be' (am/is/are) used as a separate word here. You simply put the subject (şemsiyen) and the adjective (açık) together, and the 'is' is automatically understood.

Why does the question particle take the form 'mı' here?

The question particle changes its vowel to match the last vowel of the word right before it, following 4-way vowel harmony. The last vowel in açık is ı, so the particle must be mı.