Gemara

Question: What is the source that anything within three Tefachim is Lavud (Batel to the ground)?

Answer #1 (Rav Chisda or Rav Hamnuna): It is impossible for Reshus ha'Rabim to be paved perfectly flat with a carpenter's plane.

Question: Lavud applies even high above the ground (e.g. in walls or in Sechach...)!

Answer #2: A tradition from Moshe from Sinai teaches that any gap of less than three Tefachim is Lavud (considered connected).

157a (Mishnah): A case occurred on Shabbos in which they plugged a window, and tied a jug to see whether or not a crack was a Tefach.

Sukah 18a (Abaye): If there were three Tefachim of air in the Schach, Rav Acha and Ravina argue about whether or not Lavud applies in the middle. (I.e. does it help to put Kosher or Pasul Sechach so there will not be three Tefachim together?)

The lenient opinion learns from a beam (a Korah, to permit carrying in an alley);

(Beraisa): If a beam from one wall does not reach the other wall, or if beams come out from two walls and do not meet, if the gap is less than three, one does not need another beam.

The other opinion does not learn from here. Laws of a Korah are only mid'Rabanan.

The stringent opinion learns from a Mishnah;

(Mishnah): If an Arubah (an opening between a house and the attic) is at least a Tefach [by a Tefach], if there is Tum'as Mes in the house, everything is Tamei except for under the Arubah. If there is Tum'as Mes under the Arubah, the entire house is Tahor.

The other opinion does not learn from here. Laws of Tum'ah are different. We learn them from a tradition from Moshe from Sinai.

Eruvin 25a (Rabah): If a Karfef more than 5000 square Amos was not surrounded for residence, if one made a wall [for residence] at least three Tefachim away from the wall of the Karfef, it helps. (It is considered that now it was surrounded for resident, so one may carry inside.)

This is because it is further away than the Shi'ur of Lavud.

102a (Rav Amram): If arches are within three Tefachim of each other, one may spread a mat over them [on Shabbos].

One may add to a temporary Ohel.

Rishonim

Rosh (Sukah 1:33): If there are two areas of two Tefachim each of Pasul Sechach and less than three Tefachim of air in between, it is not clear whether or not the Pasul Sechach on the two sides join.

Korban Nesan'el (400): Even though we say Lavud even when it is a stringency, we do not consider the area in between to be Pasul Sechach. Alternatively, we say Lavud to be stringent on the side. In the middle the Halachah follows Ravina to be lenient, so we do not say Lavud to be stringent.

Tosfos (Sukah 17a DH Ilu): If there are two areas of two Tefachim each of Pasul Sechach and less than three Tefachim of air in between, it is not clear whether or not the Pasul Sechach on the two sides join to four to disqualify. Air separates them, so they do not join. Or, perhaps since the separation is less than three, they join. If the total size of the Pasul Schach is less than four, surely the air does not join with it to make a Shi'ur of four, for we do not say Lavud to be stringent.

Tosfos (Eruvin 10b DH Ohr): The leather of a toilet seat joins with the gap [to be an Ohel] when the Tum'ah and the Kli [that will become Tamei] are both under the leather, like it says in Sukah, that Tum'ah in the house is not Metamei under the Arubah, and Tum'ah under the Arubah is not Metamei the house. In any case, the air joins, just like air of a Sukah joins, but one may not sleep under it. We do not say that due to Lavud, the air is considered to be sealed. A tradition from Sinai teaches that we do not say Lavud regarding Tum'ah.

Gra (in Zer Zahav on Tosefta Kelim Bava Basra 1:4): Rashi and Tosfos in Eruvin explain that the Gemara discusses joining the air for an Ohel for Tum'ah. This is difficult, for an Ohel must be a solid Tefach! Several places in Ohelos teach that air does not join for this.

Poskim

Rema (OC 502:1): If a table has walls that reach the floor, it requires a Shinuy [to erect it, e.g. to hold the tabletop in place, and put the legs under it. One may not assemble it normally.]

Beis Yosef (DH u'Mah she'Chasav Al): Tosfos permits a table with walls that do not reach to the floor. However, if they reach within three Tefachim of the floor, it looks like they touch the floor, for anything within three is Lavud.

Rebuttal (Darchei Moshe 2): We do not say Lavud to be stringent. The Rosh said so, and this is clear from Sukah 18a and the Tur (Sof Siman 632).

Magen Avraham (9): The Rema said similarly in YD 371:4 (brought below), from R. Yerucham. However, R. Yerucham did not give the reason. The Rema said that it is because we do not say Lavud to be stringent. This requires investigation. In Sukah, we say that laws of Tum'ah are different. They are a tradition from Sinai. Therefore, we do not say Lavud. Tosfos (Eruvin 10b) says that the air joins to a Tefach. This shows that we say Lavud to be stringent. However, this is only if there is a half-Tefach on each side, but not if there is a tiny amount on each side. Below (Eruvin 102a DH Lo), Tosfos says that surely spreading two strings [almost] three Tefachim apart is not forbidden due to making an Ohel [through Lavud]. This shows that even when we say Lavud, it does not make an Ohel. Really it is an Ohel for Shabbos. It permits extending an Ohel on Shabbos (102a). In any case, one may spread it on Shabbos, for he did not do anything. One may spread a half Tefach, and another half Tefach within three, even though [due to Lavud] the result is an Ohel for Tum'ah. However, a seven [and Mashehu] Tefachim wall within three of the ground is a proper wall, so one may not put a tabletop on it. Likewise, one may not place a seven Tefachim wall within three of the ground to be a Mechitzah [between a bed and Seforim] to permit marital relations (Rema 315:1). The law of erecting a table depends on the argument of whether Lavud applies to Kelim (Shabbos 8a).

Mishnah Berurah (17) and Sha'ar ha'Tziyon (17): The Magen Avraham and other Acharonim are stringent like the Beis Yosef. We do not rely on the Darchei Moshe.

Shulchan Aruch (632:4): If there are two areas of two Tefachim each of Pasul Sechach and less than three Tefachim of air in between, it is not clear whether or not the Pasul Sechach on the two sides join to four to disqualify.

Magen Avraham (5) and Mishnah Berurah (20): If the total size of the Pasul Schach is less than four, surely the air does not join with it to make a Shi'ur of four, for we do not say Lavud to be stringent (Tur). I.e. even though the two Pasul areas are considered adjacent, we do not consider [the area in between] to be sealed with Pasul Schach. See also 502:1.

Gra: This is from Tosfos 17a.

Kaf ha'Chayim (26): Tosfos and the Rosh were unsure. R. Yerucham said that it is Pasul.

Kaf ha'Chayim (27): If one cannot fix it, and he has no other Sukah, he sits in it, but does not bless.

Rema (YD 371:4): If two roofs do not touch, even though there are less than three [Tefachim] between them, they are not connected for Tum'ah. We do not say Lavud to be stringent.

Chacham Tzvi (59): The Rema erred. He overlooked Eruvin 25a. A wall within three Tefachim of the wall of the Karfef does not permit, for it is Lavud. Rashi says that it is as if it is on top of the Karfef wall. This shows that we say Lavud to be stringent, even for Isurim mid'Rabanan.

Chasam Sofer (YD 340): Tosfos and the Rosh do not say Lavud to be stringent. I.e. we do not consider the gap to be filled with Pasul [Schach] to be stringent. However, we consider the ends to be connected, whether this is a leniency or stringency. See Magen Avraham (502:9). However, in Sukah we learn from the Arubah that Lavud does not apply in the middle. Why don't we say that this is to be stringent, but we are lenient to apply Lavud in the middle?! It seems that we rely on the Seifa (Ohalos 10:4), which says that Tum'ah passes through a square Tefach to be Metamei the entire house. We don't say Lavud to be lenient. This shows that we never say Lavud in the middle.

Chasam Sofer: Rashi explained that we learn from when there is an open square Tefach. The Ritva (18a DH Girsas) explained differently. We know that a Tefach for Tum'ah is like three Tefachim elsewhere. (Lavud applies only to openings smaller than this.) Rather, the proof is when there is not a square Tefach. We do not say Lavud. However, if so, there is no proof from the Seifa (it discusses an open Tefach), so my question remains! (Perhaps we are lenient to apply Lavud in the middle!) Rather, Rashi is correct.

Chasam Sofer: The Ritva (18a DH Amar) said that Lavud does not apply in the middle. However, it applies to the Shi'ur of one's head. One may sit under such a hole. If not, the Schach would have to be as thick as a roof! I do not know his source for this Shi'ur. If so, perhaps we say similarly about Tum'ah! I.e. Lavud does not [always] apply less than three, but it applies to a small Shi'ur! However, for other matters we say Lavud [in the middle] up the Shi'ur of a head, and Tum'ah, for which the Shi'ur is a Tefach, we say it for less than a Tefach. Perhaps even when we do not say Lavud, if one is Mevatel something in the Binyan and does not want the hole, e.g. Serigei (lattice) of windows, it is considered totally closed. It is unlike an Arubah, which is made intentionally.

Note: Why does he say that people do not want the holes in Serigim? Seemingly, people use Serigim to let in light and air! It does not sound like he discusses one who used two lattices, so one will seal all the holes in the other. This requires investigation.

Sha'ar Efrayim (119): Is a lattice considered a proper Mechitzah in front of Seforim to permit relations? Seemingly, we learn from Ohalos 13:1, in which Beis Hillel say that Tum'ah enters through a lattice only if one opening itself is the Shi'ur of a drill hole. Really, this is not a proof. Regarding Tum'ah, we do not say Lavud. The Rema here connotes that we say Lavud to be lenient. He learns from R. Yerucham. R. Yerucham said Stam 'we do not say Lavud.' This implies that we do not say Lavud even to be lenient! Shabbos 157a connotes that we do not say Lavud even to be lenient. (It seems that he explains like R. Chananel, that the cracked basket was in a mound between houses. If the crack was a Tefach, Tum'ah passes to the other house.) This refers to Tum'ah. If so, regarding relations we may say Lavud. This is wrong. We need a Mechitzah due to "v'Hayah Machanecha Kodesh v'Lo Yir'eh Becha Ervas Davar". A Mechitzah through which one can see does not help.