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Determining the Techum
Outside of a City
- If a person is alone in the desert, the Techum extends 2000 amot beyond the 4 amot area him. If one is in a house, the Techum is 2000 amot from outside the house. 
- If a person starts Shabbat outside a city he can determine the directions of his techum and have 2000 amot in each of those directions. He also is allowed to travel until the end of 2800 amot in the direction of the corners of the square he choose. However, it is only allowed to carry until the end of 2800 amot if he decided on his square.
- If one's Techum includes the entire length of a city, which is defined above, the city is only considered 4 amot and one may walk beyond that city for the rest of one's 2000 amot. For example, if one's dwelling place is 500 amot from a city and the city is 1000 amot long, the city is only considered 4 amot. Therefore, one is able to walk another 1496 amot after the city. However, if one's Techum ends in the middle of the city, the city is not considered 4 amot and one may not walk beyond one's Techum.
Techum of a City
- If one is in a village, town, or city, one may have 2000 amot from the outer bounds of the city, depending on the density of the houses. If there are 6 homes each within 70.66 amot of another home are considered a city and the Techum would begin from 2000 amot outside the group of houses. Any home that is within 70.66 amot of the established city is included in the city, however, a house which is more than 70.66 amot from other houses is not included in the city and the Techum for residents of that house is only 2000 amot from the edge of that house. Therefore, a suburban area with houses separated more than 70.66 amot are not considered part of a city and residents of a house only have 2000 amot from that house. 
- Many cities have 2000 from the edge of the city which are drawn as a rectangle along the directions of a compass, from the edge of the northern most house, eastern most house, southern most house, and western most house. However, if the city is already rectangle, L-shaped, or arc shaped may not have this extension of squaring off the city.
- Because the laws of establishing a Techum and extending the Techum with a Eruv Techumin are complicated one should consult a local Orthodox Rabbi. 
- If two cities are within 141.3 amot of each other they are considered like one city and the techum is drawn around both of them.
- If when drawing the square around a city that would include another city the two cities are considered one large city and the techum is drawn around both of them.
Walking to the edge of the Techum
- It is forbidden to walk to the edge of the Techum in order to leave on a journey quickly after Shabbat. However, if the action one is going to do after Shabbat could theoretically have been done on Shabbat, it is not forbidden to walk tot the edge of the Techum waiting for the end of Shabbat. For example, one may walk to the edge of the Techum in order to bring one's animal back because theoretically one could have done this on Shabbat if there were houses extending the Techum. Also one may walk to the edge of the Techum in order to collect fruit which fell before Shabbat and aren't Muktzeh because theoretically one could have done so on Shabbat if there were walls surrounding the path (which would permit carrying on Shabbat). 
- One may walk to one's garden within the Techum in order to pick fruit after Shabbat since it is not evident that one is walking there for that purpose.
Techum above Ten Tefachim
- There is an unresolved discussion in the Gemara whether there is techum above ten tefachim. We are lenient if a person started Shabbat above ten tefachim that he doesn't have a techum of two thousand amot as long as he didn't yet land or reach a place that is connected to the ground. This is relevant to a person who was on a boat at the beginning of Shabbat.
- A wagon is considered within ten tefachim of the ground since the wheels rest on the ground and the wagon itself if bigger than 4 tefachim.
Techum of Property
- A barrel that belongs to two people that was split up on Yom Tov, each part has the techum of the owner of that half even though it was only split up on Yom Tov.
- See a summary of the halacha's of Techum with pictures on TechumShabbos.com
- Shulchan Aruch O.C. 396:1, 495:1, 611:1. Peninei Halacha discusses why the allowance for ochel nefesh doesn't permit going beyond the techum on Yom Tov even for ochel nefesh.
- Shulchan Aruch 397:1, Kitzur Shulchan Aruch 95:1. Mishna Brurah 397:1 writes that some hold that walking beyond 12 mil on shabbat is a Biblical prohibition, while others consider it a rabbinic prohibition.
- 39 Melachos (Rabbi Ribiat, vol 4, pg 1386-7)
- Chazon Ish 110:24 based on Gra 345:1, Mishna Brurah 345:5. This is the opinion of the Rambam. According to the Rabbenu Tam (Eruvin 51a, Yoma 67a a person always can travel until 2800 amot even without choosing the square. However, Rashbam held that one's square is automatically determined to correspond with the north-south directions of the earth and then one can walk until the end of those corners. Tosfot Yoma and Rosh Eruvin reject Rabbenu Tam.
- Kitzur Shulchan Aruch 95:12
- Kitzur Shulchan Aruch 95:2
- 39 Melachos (Rabbi Ribiat, vol 4, pg 1387-9). Rabbi Ribiat quotes the Minchat Shabbat who explicitly writes that we do have the halachic category of chatzerot nowadays in order to form a city. We still require that there's 2 houses to create a chetzer but if there's 6 houses we have a city. Torat Shlomo Eruvin 19:19 p. 144 explicitly writes that even though we don't use the chatzerot today like rooms you can create a city without chatzerot. (See Chazon Ish OC 110:20 who might support this.) However, Rav Hershel Schachter (Shabbat Shiur 99 (very end) and Shabbat Shiur 100 (beginning)) holds that nowadays we don't have a city for techum since we don't use our chatzerot today like a room of the house. Therefore, since we need 3 chatzerot to form a city (S"A 398:10) we can't have a city today. This is based on the Rama, Chazon Ish OC 65:52, and is discussed by Imrei Baruch Eruvin 40-41. The definition of a city is relevant to techum and eruv. See, however, Magen Avraham 398:15 who implies that an established city doesn't need 3 chatzerot.
- 39 Melachos (Rabbi Ribiat, vol 4, pg 1391-2)
- 39 Melachos (Rabbi Ribiat, vol 4, pg 1393)
- Shulchan Aruch O.C. 398:7
- Minchat Yitzchak 8:33 explains that since the squaring of a city is taken into account before we add the 70 amot of the city if the squaring off of a city includes another city everything should be considered one large city. His proof is the concept of Eruvin 55a that the arms of a bow-shaped city join as long as they are within 4000 amot of each other. The Chazon Ish 110:16 wasn't sure whether or not the cities join when the squaring off of each other join. Rav Chaim Kanievsky in Shoneh Halachot 398:19 and 21 writes that it seems that his conclusion was that they do join. Zecher Tzvi (Techum Shabbat p. 21) agrees. Dirshu 398:21 cites Rav Wosner (Kitzur Hilchot Medidat Techumin) as agreeing but Rav Elyashiv as holding that one should be strict. Mechzeh Eliyahu 1:74 and 77 is lenient. Machneh Yisrael of Rabbi Dimitrovsky p. 19-22 writes that one who is lenient has what to rely upon and supports this from the Rambam and Meiri. He also cites Rav Elyashiv as being strict.
- Kitzur Shulchan Aruch 90:3
- Kitzur Shulchan Aruch 90:3
- Gemara Eruvin 43a-b, Shulchan Aruch OC 404:1
- Shulchan Aruch 404:1
- Shoel Umeishiv 5:3
- The Gemara Beitzah 37b records a dispute between Rav and Shmuel whether we hold of the concept of berierah. Berierah is that we can view an eventual decision as though it already happened to clarify what is the case now. Rav holds of it and Shmuel does not. The gemara’s conclusion (38a) is that for derabbanan concepts we hold of Berierah. This is codified by the Rambam (Yom Tov 5:20) and Shulchan Aruch 397:10.