The Laws of Prayer (The morning services) (43)

1. One who is sitting in his house when another person comes and starts praying within four cubits of him, he is not obligated to arise as the second person has come within his borders. However, a man of piety should stand or sit further away. (Shulchan Aruch and Mishne Berurah 102)

2. However, one who is sitting in the synagogue when another person comes and starts praying within four cubits of him, is obligated to stand up, as he is in a place designated for prayer. One should be stringent about this even if each person has a fixed place which he has paid for. However, one who is learning Torah does not need to be stringent in this case. (ibid)