Here, I will be discussing my perspective on the motivation for Topology. The following information comes from my notes on Topology found here.
We are motivated by the following definitions and theorems:
- Definition. (Open and Closed) A set is open if such that (interior points). A set is closed if is open.
- Ex) (a) Let . Then to be an open set, such that . However, if we chose or for , there is no in . Hence, is not an open set. On the other hand, we define , which is open. Therefore, by the definition of a closed set, is a closed set. (b) Let . Then we take , such that . We can see that all points contained in have a -interval. I.e., for . Therefore, is an open set.
- Theorem. (Open Sets) (a) If is a collection of open sets, then is open. (b) If is a finite collection of open sets, then is open. (c) Both and are open.
- Theorem. (Closed Sets) (a) If is a collection of closed sets, then is closed. (b) If is a finite collection of closed sets, then is closed. (c) Both and are closed.
- Definition. (Limit Points) is a limit point of if , the .
From this, we can form our definition of a topology.