Things That Are Invalid For S'chach, having four Seifim: 1. Invalid s'chach, makes a sukkah invalid in the middle if it is four tefachim. However, less than 4 is valid and one is permitted to sleep under it. From the side, it does not invalidate except if the invalid s'chach measures 4 amot, but less than 4 it is valid, for we say it is a curved wall; for we could say that we see it as if the wall is bent, and we consider this invalid s'chach to be from the body of the wall. This is a law given to Moshe at Sinai. Therefore, for this reason, a house which is reduced in the middle and one covers the middle area with s'chach, and there remains from the ceiling around the valid s'chach less than 4 amot between it and the walls, it is valid. However, we do not sleep underneath it whenever it measures 4 tefachim. With regard to what have these things been said? With regard to a large sukkah that has, in addition to the invalid s'chach, more than 7 x 7 tefachim. However, in a small sukkah which only has 7 x 7, whether in the middle or on the side, when there are three tefachim, it is invalid. When there are less than three tefachim, it is a valid sukkah and we may sleep underneath it and it joins together to complete the measure of the sukkah.
2. Open space, whether great or small, is the same. Whether in the middle or on the side within three tefachim it is invalid. If it is less than three tefachim it is valid, and it joins together to complete the sukkah and we do not sleep underneath it. RAMA: And this is specifically when it goes along the entire face of the sukkah (Rabbeinu Yerucham), or it has enough space for one to fit their head and most of their body, but without this it is permitted, because there is no sukkah that doesn't have holes (R"an, end of first perek). This, that invalid s'chach invalidates a sukkah with four, and open space within three, is specifically when the sukkah is split in two, and there does not remain a valid measure for a kosher sukkah with walls in any one place. But if there does remain a valid measure for a kosher sukkah in one place - this place is kosher. And even outside if they are joined from the side (Tur).
3. Invalid s'chach less than four and open space next to it less than three do not join together to invalidate (the sukkah). For this reason, if an open space is three in one location and even a small part of this open space is invalid s'chach, it (the sukkah) is valid. This applies to a big sukkah. However, in the case of a small sukkah that only measures 7 x 7, if there is three tefachim between the two of them, they join together to invalidate the sukkah.
4. If there are 2 tefachim of invalid s'chach and 2 other tefachim of invalid s'chach, and an open space less than three interrupts between them, there is reason to doubt if two tefachim of the invalid s'chach join together to invalidate the sukkah.