Gemara

5a: Rav Nachman cited a tradition that a Lechi or Korah permits a Mavoy only if its length exceeds its width, and houses and Chatzeros are open to the Mavoy.

73a (Mishnah): If five Chatzeros open to each other and to a Mavoy... if they were Mishtatef in the Mavoy, everything is permitted.

73b (Rav): The text of the Mishnah does not say that the Chatzeros open to each other.

Question (Abaye): What forced Rav to say so?

Answer: Rav holds that a Lechi or Korah permits a Mavoy only if houses and Chatzeros are open to it.

(Shmuel): Even if one house and one Chatzer open to it [a Lechi or Korah permits it];

(R. Yochanan): Even a ruin [and a Chatzer with a house suffice].

Question (Abaye): Does R. Yochanan say that even a path in a vineyard suffices?

(Rav Yosef): No. He requires [at least] a ruin, for it can be used for Dirah. A path cannot.

Question (against Shmuel - R. Elazar - Mishnah): A Mavoy to Chatzeros is like a Chatzer to houses. (A Lechi or Korah permits a Mavoy only if Chatzeros (i.e. at least two) open to it, and houses open to each Chatzer.)

Shmuel did not answer.

He accepted R. Elazar's words [and retracted, and held like Rav];

A case occurred in which Shmuel approved of a Lechi for a Mavoy. [After Shmuel died,] Rav Anan knocked down the Lechi. (A house open to one of the Chatzeros lost its resident, so only one proper Chatzer opened to the Mavoy.)

Shabbos 131a - Question (Rav Chanina Chuza'ah): Rav should forbid if the houses did not make Eiruvei Chatzeros, since one may not transfer from the houses to the Chatzeros. It is as if the houses are sealed, and only Chatzeros open to the Mavuy!

Answer (Rava): All the houses in each Chatzer can be Mevatel (pardon) their share in the Chatzer to one house, permitting that house to transfer to the Chatzer. (It is as if the Chatzer belongs exclusively to that house.)

Question: Still, it is as if there is only one house in each Chatzer!

Rishonim

Rif (23a) and Rosh (6:19,20): Rav holds that the Chatzeros cannot open to each other, for a Lechi or Korah permits a Mavoy only if houses and Chatzeros open to it. If they are open to each other, they are like one Chatzer. The Halachah follows Rav. Firstly, Shmuel retracted to agree with Rav. They are two against R. Yochanan. Also, the Mishnah supports Rav, and Rav Nachman holds like him.

Rambam (Hilchos Eruvin 17:8): If there are not houses and Chatzeros open to a Mavoy, e.g. there is only one house or only one Chatzer, it is permitted only through two Lechiyim or a plank four wide.

Rosh (20): Maharam says that even though the Halachah is like Rav, that houses and Chatzeros must open to the Mavoy, but not like Rav's opinion that if the Chatzeros open to each other, they are considered like one. We rule like the Stam Mishnah, which says that they open [to each other]. Also, it says that Rav taught that the text does not say open. This implies that our text says open.

Terumas ha'Deshen (74): Our custom is to be Machshir Mavo'os through Tzuras ha'Pesach, without a Lechi or Korah. This is proper, for nowadays most cities with a wall were settled and later surrounded. If so, the entire city is like a public Chatzer without an Eruv. If so, one must fix them the way we do when we come to divide a Chatzer into three or four domains. If the Chatzer had small paths between houses, we need Tzuras ha'Pesach or two planks of any size (Hagahos Maimoniyos citing Maharam).

Poskim

Shulchan Aruch (OC 363:26): A Lechi or Korah permits a Mavoy only if [at least] two Chatzeros open to it, and [at least] two houses open to each Chatzer.

Beis Yosef (DH v'Eino): Even though on 12b, Rashi explained that two Chatzeros must open to the Mavoy, and one house to each Chatzer, Tosfos and the Rosh (1:17) said that what he wrote on 74a is correct, that two houses must open to each Chatzer. Semag and Semak agree, and the Yerushalmi says so. The Mordechai says that if it does not have Chatzeros and houses open to it, it requires a plank four wide, or two planks of any width, or Tzuras ha'Pesach. This is correct. We said only that it is not permitted through a Lechi or Korah, but other ways to be Machshir a Mavoy work.

Beis Yosef (DH v'Chasav): The Rashba (1:601) was asked about a Mavoy that cannot be permitted through a Lechi or Korah. He answered that Tzuras ha'Pesach is a full Mechitzah. If it has two walls, Tzuras ha'Pesach is a third wall, so a Lechi or Korah suffices for the fourth side. The Rambam said that if there are not houses and Chatzeros, e.g. there is only one house or only one Chatzer, it is permitted only through two Lechiyim or a plank four wide. The Magid Mishneh struggled to explain this, and concluded that this is if there is only one house, even if there are many Chatzeros, or if there is only one Chatzer, even if there are many houses. The house open directly to the Mavoy. However, Rashi said that there are two houses in each Chatzer. The Rashba agreed. Since the Rosh and many Gedolim and the Yerushalmi are like Rashi, we hold like him.

Beis Yosef (DH Kasav Rabbeinu): The Rambam (1:19) holds that we rely on Shituf in place of an Eruv. We must say that houses and Chatzeros must open to the Mavoy, for if there is less, it is only like a Chatzer. The Rashba (5:203) says so. The more that it is for residence and private usage, it needs fuller walls. Therefore, Chatzeros that house-owners use for more private matters and to eat there, require fuller walls. The same applies to Mavo'os without houses and Chatzeros open to the Mavoy. There are few residents, so they are not ashamed to eat there and use them, so they are like a Chatzer. More residents diminish usage, so any Mechitzah suffices.

Magen Avraham (25): If there are not many residents, they use it for private matters, so we require more Mechitzos. If so, we may be lenient about our Mavo'os, since there are many residents. However, perhaps we do not distinguish in a Mavoy.

Gra (DH u'Shnei): Shabbos 131a proves that two houses must open to each Chatzer.

Bi'ur Halachah (DH Shnei): It does not matter whether the Chatzeros are on one side or opposite sides of the Mavoy, or on the middle wall.

Rema: This is even if all the Chatzeros are open to each other. Some disagree.

Beis Yosef (DH u'Mah she'Chasav Rabeinu): The Rif brought Rav's law. This connotes that he holds that the Halachah follows him. It seems that the Rambam (5:23) agrees. We hold like them.

Note: Perush ha'Mishnayos clearly explains like Rav, but I do not see any connotation from Halachah 5:23 like Rav.

Taz (17): The Rif and Rambam hold that since they open to each other, it is as if they are one Chatzer, and we require two Chatzeros. The Rosh, Tur and Mordechai hold like the first opinion.

Magen Avraham (26): Since they can be Me'arev through the openings, they are considered like one Chatzer. The same applies to houses open to each other.

Mishnah Berurah (103): Acharonim says that since this is a mid'Rabanan matter, we are lenient.

Shulchan Aruch (ibid.): If it is lacking any of these conditions, it is permitted only through a plank four wide or two planks of any width, or Tzuras ha'Pesach.

Rema: Nowadays the custom is to fix all Mavo'os through Tzuras ha'Pesach, for all our Mavo'os are considered Chatzeros. The custom is to use a rope tied to the width of the Mavoy. It is not wide enough to be considered a Korah. Rather, it is Tzuras ha'Pesach. There must be two beams 10 Tefachim tall directly under the rope. This helps even for a Mavoy open on both ends, or even for a Chatzer, and whenever Tzuras ha'Pesach helps.

Magen Avraham (28): The walls of the Mavoy do not count like beams. If they did, why did Chazal require a Korah a Tefach thick? Any amount should suffice [if it rests on the walls, for it is Tzuras ha'Pesach]! Why is this unlike Sukah? There, Tzuras ha'Pesach may rest in the walls! Perhaps this is a mere stringency.

Gra (DH Eino): Whenever it cannot be permitted like a Mavoy, it has the law of a Chatzer.

Mishnah Berurah (111): This is because our Mavo'os do not have houses and Chatzeros open to the Mavoy. Even though a plank four wide or two thin planks suffices, the custom is to use Tzuras ha'Pesach due to a Mavoy Mefulash, or a breach more than 10 Amos, which require Tzuras ha'Pesach.

Mishnah Berurah (114): If it is even a drop to the side, it is Pasul.