Gemara

(Rav Gidal): If the interior of a house is less than 10 tall, one may not carry four Amos inside.

(Abaye): If one dug out a cavity four by four in the floor, and now there it is 10 tall, one may carry in the entire house.

(The place of the cavity is a proper Reshus ha'Yachid, even if it is not adjacent to the walls.) The rest of the house is like Chorei [holes coming out from] Reshus ha'Yachid;

Chorei Reshus ha'Yachid are like Reshus ha'Yachid;

(Abaye): Chorei Reshus ha'Rabim are like Reshus ha'Rabim;

(Rava): They are not like Reshus ha'Rabim.

Question (Rava, against Abaye): Rav Dimi taught that 'Karmelis' (in the Beraisa 6a) includes a corner next to Reshus ha'Rabim, even though it is Chorei Reshus ha'Rabim!

Answer (Abaye): This is because it is inconvenient to use it (walk there). Regular Chorei Reshus ha'Rabim are convenient to use.

Question (against Abaye - Mishnah): If one threw something four Amos at a wall:

If this was above 10, he is (exempt) like one who threw in the air. If this was below 10, he is (liable) like one who threw on the ground.

Question: When it was below 10, why is he liable? It did not rest [outside of four Amos, for it bounces back]!

Answer (R. Yochanan): The case is, he threw a fat date. (It stuck to the wall.)

Why didn't R. Yochanan say that the object landed in a hole in the wall, i.e. Chorei Reshus ha'Rabim!

Answer #1 (Abaye): An object would probably bounce off and not land in the hole.

Answer #2 (Abaye): The Mishnah cannot discuss a wall with a hole, for then the Reisha would not say that above 10 is like throwing in the air! [It is Reshus ha'Yachid. He is liable!]

Suggestion: Perhaps the hole is not four by four!

Rejection: Rav Yehudah taught that R. Meir and Chachamim argue about one threw above 10 and it landed in a hole Kolshehu (less than four);

R. Meir says Chokekin Lehashlim. (We view it as if it was carved out. It is as if the hole is four by four);

Chachamim do not say Chokekin.

Rishonim

Rif and Rosh (2b and 1:12): The Halachah follows Rava. Chorei Reshus ha'Rabim are unlike Reshus ha'Rabim. Therefore, if they are four by four Tefachim wide and 10 Tefachim tall, they are Reshus ha'Yachid. If they are four by four but less than 10 tall, they are Karmelis. If they are less than four by four, even if they are 10 tall, they are Mekom Petur.

Ran (DH Chorei): Chorei Reshus ha'Rabim are holes in the wall on the side of Reshus ha'Rabim. They are not like Reshus ha'Rabim. They are a Reshus unto themselves. This is when they are above three Tefachim.

Rambam (Hilchos Shabbos 14:10): Chorei Reshus ha'Yachid are like Reshus ha'Yachid. Chorei Reshus ha'Rabim are unlike Reshus ha'Rabim. They are judged according to their size. If a hole on the side of Reshus ha'Rabim is four by four wide and 10 tall, it is Reshus ha'Yachid. If it is less than 10 tall, it is Karmelis. If it is less than four by four, it is Mekom Petur. This is if it is at least three high. Anything less than three is like the ground.

Magid Mishneh: The Rashba says that Chorei Reshus ha'Yachid are holes of Reshus ha'Yachid that go through towards the inside. Chorei Karmelis are judged according to their size, like Chorei Reshus ha'Rabim.

Tosfos (7b DH v'Halchah): The Ri says that if a hole is Mefulash (passes totally through the wall), and on the inside (Reshus ha'Yachid) it is four wide, but on the outside (Reshus ha'Rabim) it is not four, all agree that it is Chorei Reshus ha'Yachid. R. Meir and Rabanan argue about whether we say Chokekin to consider the place to be four by four. If the hole is truly Kolshehu (less than four everywhere), even R. Meir does not say Chokekin. The Yerushalmi connotes like this. It says 'according to R. Meir, whether or not there is four, we view the wall as if it were cut. I.e. if it is above 10, it is considered Chorei Reshus ha'Yachid. Below 10, Bnei (people of) Reshus ha'Yachid do not use it, for Bnei Reshus ha'Rabim use it.' Our Gemara suggested that even though Chorei Reshus ha'Yachid applies even if it is less than four wide, it is like throwing in the air because it does not land on a place four wide. It rejected this due to Rav Yehudah's teaching. We do not say that the Mishnah literally discusses a hole Kolshehu, and then no one says Chokekin, for Stam holes are like this. (I.e. they are four wide on one side - Maharsha.) Also, we needed to say that it discusses a fat date. If it discussed a hole Kolshehu, and we do not say Chokekin, why is below 10 like throwing on the ground? It did not land on a place four wide!

Rashba (Avodas ha'Kodesh): Chorei Reshus ha'Yachid are holes in walls of Reshus ha'Yachid that open to the inside. If they open only to the outside, to Reshus ha'Rabim or Karmelis, they are not Reshus (ha'Rabim - the Beis Yosef says that this is a printing mistake. It should say "ha'Yachid"). Rather, they are judged based on their size.

Poskim

Shulchan Aruch (OC 345:4): Chorei Reshus ha'Yachid that face Reshus ha'Yachid are like Reshus ha'Yachid.

Magen Avraham (2): Tosfos said that below 10 they pertain to Reshus ha'Rabim. This was only the Hava Amina, unlike Olas Shabbos.

Eshel Avraham: Tosfos said so according to Abaye, but not according to Rava. If the hole is Mefulash, below 10 it is like Reshus ha'Rabim, like in Sa'if 13.

Eliyahu Rabah (8): The Rashba says "even if they are below 10 and they are not four by four" (they are Reshus ha'Yachid). This is like the Magen Avraham. R. Yerucham says in the name of Tosfos that if they pass from Reshus ha'Yachid to Reshus ha'Rabim, below 10 they are like Chorei Reshus ha'Rabim, and above 10 they are Chorei Reshus ha'Yachid. The Magen Avraham and Olas Shabbos overlooked this (that they argue like the Rashba and R. Yerucham).

Kaf ha'Chayim (17): Shulchan Shel Arba says that the Rashba does not support the Magen Avraham. The Rashba discusses a hole open only to Reshus ha'Yachid!

Mishnah Berurah (9): Even if the hole faces also Reshus ha'Rabim, since it is Mefulash and enters also Reshus ha'Yachid, one may not take anything from the hole out to Reshus ha'Rabim or Karmelis next to it. If one put from Reshus ha'Rabim into it, he is liable. Tosfos obligates only if the hole is four by four. However, the Rashba and Me'iri say that Chorei Reshus ha'Yachid are like Reshus ha'Yachid even if they are not four wide.

Rema: If they face outside and do not pass to the inside, they are judged based on their size, like below (Sa'if 13).

Eshel Avraham (3): The Rema connotes that if they are Mefulash, they are Reshus ha'Yachid regardless of their size. The Mechaber was not precise when he said that they open to Reshus ha'Yachid. Really, they open to both sides.

Gra (DH v'Im): The Rashba discusses only holes above 10 that are not four wide, even according to Rabanan, for we hold like Rav Chisda, who does not require Hanachah on a place four wide in Reshus ha'Yachid. Below 10 they are holes of Reshus ha'Rabim, like Tosfos says. The Magen Avraham is very difficult. See Tosfos Eruvin 98b DH u'Mishtamesh. (He explains the Gemara there to say that if a window opens to Reshus ha'Rabim, one in the house may use holes in the wall only above 10 Tefachim.)

Damesek Eliezer: The Rema understands that the Rashba discusses even if the hole is below 10. If it opens to the inside, it is Reshus ha'Yachid. The Gra disagrees. The Rashba discusses only holes above 10. He teaches that even though it is not four wide on the inside, and we hold like Rabanan of R. Meir, who do not say Chokekin, even so one who throws there (from Reshus ha'Rabim) is liable, since we hold like Rav Chisda. Therefore, whenever it is open to the inside, it is like Reshus ha'Yachid. However, if it is below 10 and Mefulash, it is a Karmelis if it is four by four, and if not it is Mekom Petur, like Tosfos says. The Magen Avraham said that this was the Hava Amina. I.e. the Beraisa said that throwing above 10 is like throwing in the air, and below 10 is like throwing on the ground. R. Yochanan answered that he threw a fate date. Tosfos challenged the Hava Amina that he threw into a hole. Presumably, above 10 and below 10 are similar, i.e. in both cases the hole is Mefulash. If so, why is below 10 like throwing on the ground? Also this is Chorei Reshus ha'Yachid! (One who threw from Reshus ha'Rabim is liable for even less than four Amos - PF.) This forced Tosfos to say that Bnei Reshus ha'Yachid do not use it below 10. All this was in the Hava Amina, that he threw into a hole. We concluded that the date stuck to the side, so we can say that also below 10 is Chorei Reshus ha'Yachid if it opens also to the inside. The Gra totally rejected this. If there is a reasoning to say that below 10 is Chorei Reshus ha'Yachid when it is Mefulash, the Gemara should have said that we cannot discuss throwing into such a hole, for then below 10 is unlike throwing on the ground! Since the Gemara did not say so, this shows that below 10 is unlike Reshus ha'Yachid.

Damesek Eliezer: I say that the Magen Avraham means that Tosfos said so according to the Hava Amina that Chorei Reshus ha'Rabim are like Reshus ha'Rabim. If so, usage of Bnei Reshus ha'Rabim is Mevatel usage of Bnei Reshus ha'Yachid. We concluded that Chorei Reshus ha'Rabim are unlike Reshus ha'Rabim. Usage of Bnei Reshus ha'Rabim is not significant. Therefore, any Mefulash hole is like Reshus ha'Yachid, because Bnei Reshus ha'Yachid use it. Tosfos in Eruvin holds like they said in Shabbos, that Stam holes are made Mefulash, and they are four wide only on the inside. Therefore, [from the house] one may use only the holes above 10. Holes below 10 are not Reshus ha'Yachid. If we say Chokekin, they are Karmelis. Korban Nesan'el on the Rosh in Eruvin said so, and challenged the Magen Avraham. In 353:4, the Gra proved that the Mechaber holds that holes below 10 are unlike Reshus ha'Yachid, unlike the Rema and Magen Avraham.

Mishnah Berurah (10): The Mechaber connotes that they are like Reshus ha'Yachid even if they are below 10. Since they are Mefulash to the inside, they are like Reshus ha'Yachid. The Rema connotes like this. He says that if they do not pass to the inside, they are judged according to their height and width. This connotes if they are Mefulash, they are like Reshus ha'Yachid regardless of their height and width. The Magen Avraham and Tosefes Shabbos say so. Eliyahu Rabah says that Rishonim argue about this. Some say that since it is below 10, people of Reshus ha'Yachid do not use them, because they are low and people of Reshus ha'Rabim use them. Therefore, they are like Chorei Reshus ha'Rabim. The Gra holds like this.

Kaf ha'Chayim (18): A hole in a wall between two Chatzeros is like air accessible to both of them. Neither may use it without an Eruv.

Kaf ha'Chayim (19): If a tree is 10 tall and four wide, and a basket hangs on it, it is like Chorei Reshus ha'Yachid. Even if the bottom of the basket is three above the ground, and goats could stick their heads under it, this does not affect the law of Chorei Reshus ha'Yachid.

Shulchan Aruch (13): Holes in walls facing Reshus ha'Rabim, if they are at least three high, they are judged according to their size.

Beis Yosef (DH Chorei): The Ran says that we discuss holes above three. If they are within three, they are Reshus ha'Rabim.

Mishnah Berurah (46): If they are below three, even if they are four by four wide, they are considered Reshus ha'Rabim.