How is the verb "to open" written in korean?

Why is the verb “to open” written as 열어요 and as 열지요 in this lesson?
Any clues?

To my ears it’s just a bit more of an indirect and polite way to request someone to do something.

Example: “밥을 좀 드세요” vs. “밥을 좀 드시죠?”, where the first is a direct command and the second is kind of an indirect suggestion.

Thanks Alex.