Suppose $G_i$ is closed set, how to prove $\cup_{i=1}^n G_i$ is closed.
Please do not use $(\cup_{i=1}^n G_i)^c = \cap_{i=1}^n ({G_i}^c)$. That is you could not use the theorem that a set is closed if and only if the complement of the set is open.
The question is derived from Rudin's PMA. The definitions are
(a) If every point of a set is interior point, then the set is open.
(b) If every limit point of a set is in the set, then the set is closed.
Becase the caption of the chapter is topology, so I use the tag general-topology. However, The book have not define topology or order topology.
Sorry for my bad English, I didn't mean to offend you by saying that.