Rif Eruvin 3a {Eruvin 11b continues ... 12b}

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3a

{Eruvin 11b continues}
Do they {the boards on the side and the board on top} need to touch or do they not need to touch?
To explain, these boards which are on the sides of the alleyway, if they are 10 handbreadths high and to not reach the korah on top, what?
Rav Nachman said: They do not need to touch.
And Rav Sheshet said: They do need to touch.

And the halacha is like Rav Nachman for there is a brayta like him. What is this? That they learnt {in a brayta}: a kippa {a rounded arch}- Rabbi Meir obligates it in mezuza and the Sages exempt it. And they are alike in the case where there is in its feet {a vertical part of the arch} ten handbreadths, that it is obligated.

Thus, it is clear that when there is 10 {handbreadths} in its feet, even though the ceiling at the top part of the arch does not touch the feet, it is an entrance, and is obligated in mezuza. So too here, since there is ten {handbreadths} in the side boards, even though they do not reach {/touch} the ceiling, it is an entrance.

MISHNA:
To legalize {the carrying within} an alleyway:
Bet Shammai hold {that you need} both a lechi and a korah.
And Bet Hillel hold either lechi or a korah.
Rabbi Eliezer said: Two lechis.

In the name of Rabbi Shimon {our gemara: Rabbu Yishmael} a disciple stated before Rabbi Akiva: Bet Shammai and Bet Hillel do not argue regarding an alleyway which is less than 4 cubits that it may be permitted via a lechi or a korah. Regarding what do they argue? Regarding that which is from 4 cubits until 10 - that Bet Shammai say a lechi and a korah, and Bet Hillel say a lechi or a korah.

Rabbi Akiva said: On this and that they argued.

Gemara:
Like whom {was the Mishna}? It is not like Chanania {who recorded a different dispute between Bet Shammai and Bet Hilllel} and not the Tanna Kamma {who said tzurat hapetach on one side and a lechi and korah on the other}.
Rav Yehuda said: This is what it means to say: To legalize {the carrying within} a closed alleyway, howso? Bet Shammai say a lechi and a korah, etc., and the dispute between Chanania and the Tanna Kamma is in regard to an open alleyway.

Rabbi Assi cited Rabbi Yochanan: A courtyard {which is breached to a place forbidden to it} requires two strips from two sides, each strip any width at all, and if only from a single side, you need the strip to be four handbreadths.

{To explain: if a courtyard which is breached in the entirety of one wall {=bemilu`o} to a place which is forbidden to it, say a public domain, it is forbidden to carry in that courtyard. However, if some of the breached wall remains standing, and there are less than ten cubits of breach, then one can treat this missing wall as an entrance, even without placing a korah on top. How so? If there is a strip of wall standing to the left and the right of the breach (or one erects strips there}, even of any width whatsoever, it is reckoned an entrance and it permitted to carry in the courtyard. If there is only a a portion of the breached wall remains standing on the left or the right side, then this strip must be some significant length = 4 handbreadths.}

The Sages learnt {in a brayta}: If the arm of the sea protruded into a courtyard, they may not draw from there on Shabbat unless they made a partition 10 handbreadths high.
With regard to what are these words said? Where the breach was greater than 10 cubits {wide} but 10 {or less} he needs nothing.
"Drawing" is what he may not do, implying that he may carry {his own items in the courtyard}! But is the courtyard not breached to an area forbidden to it? With what are we dealing with?
{Eruvin 12b} Where there are banks {on the side of the arm of the sea, such that it is not breached in its entirety}, which are like strips {pasim}, and these strips which are on the two sides and higher than the sea make carrying permissible in the courtyard, but a partition needs be made within the water to allow taking and drawing the water from it.

Rav Yehuda cited Rav: An alleyway whose length equals its width is not permitted by a lechi of minimal measure.
Rav Chiyya bar Ashi cited Rav: An alleyway whose length equals its width is not permitted by a korah a handbreadth thick.
Rabbi Zera said: How well the words of the elders agree! Since the length is as the width, it is reckoned a courtyard {rather than an alleyway} and a courtyard is not permitted via a lechi or korah but rather with a strip of 4 handbreadths {if on only one side of the breach, or any measure at all if on two sides of the breach}.

Rav Nachman said: We have a tradition: What is an alleyway that is permitted via a lechi or korah?