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.

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