A phrase consisting of a -고 form and a form of 있- is, stays corresponds to English verb phrases like is writing, is eating, is buying, as opposed to simple forms like writes, eats, buys.
- 나는 넥타이를 매고 있어요.
I am wearing a necktie. - 모자를 쓰고 있습니다.
I am wearing a hat.
This correspondence is by no means 100%, however; we say is going where the Korean uses the single verb 가요. For most cases of -고 있어요 you can substitute a simple verb, so that (e.g.) 먹어요 eats covers about the same ground as 먹고 있어요 is eating. This does not work in the other direction, though. You may not automatically substitute a -고 있어요 phrase for every simple verb form.
The -고 of this pattern behaves in the same way as the -고 of -고 싶어요. In other words, it keeps the ㄹ of ㄹ-extending verbs: 살고 있어요 is living. Insofar as this pattern emphasizes that a particular action or process is in progress, it can only be used with processive bases.
To show tense, you change the verb 있어요: 쓰고 있어요 is writing becomes 쓰고 있었어요 was writing. For other sentence types you also make the change on the last verb, as you can see in 쓰고 있지만 is writing, but. . .
For honorific expressions you substitute 계세요 for 있어요, so that 쓰고 있어요 becomes 쓰고 계세요 (or 쓰시고 계세요) to mean [someone esteemed] is writing.
But the negative can be made either on the underlying verb or on the expression as a whole, with slightly different meanings. If you merely want to deny an assertion you can say 쓰고 있지 않아요 is not writing, but if you want to imply that the subject goes along nicely without the activity you say 쓰지 않고 있어요 is not writing [for the time being, these days, etc. ] or gets along (manages to get by) without writing or keeps away from one’s typewriter. Both kinds of negative can be put into the various other sentence types, so that for the meaning isn’t writing, but. . . you will hear both 쓰고 있지 않지만 (or 쓰고 있지 못하지만 ) — usually as a denial of an assertion — and 쓰지 않고 있지만 (or 쓰지 못하고 있지만 ). Here are some more examples.
- 어머니는 아침을 아직 잡수시고 계세요.
Mother is still eating breakfast. - 고모가 밖에서 기다리고 있으니까, 빨리 해요.
Auntie (father’s sister) is waiting [for you] outside, so hurry up. - 동생이 아직 자고 있어요.
My little brother (or little sister) is still asleep. - 시어머니께서 새 옷을 입고 계셨어요.
The mother-in-law was wearing [or had on] a new dress. - 큰아버지는 요즘도 술을 드시지 않고 계십니다.
My uncle is keeping off the booze lately (still/too). - 이모는 아직 렌즈를 찾지 못하고 있어요.
Auntie (mother’s sister) still can’t find her contact lens.