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**Closed and open at the same time??!**

I am doing a reading course with a professor using Rudin's Real and Complex Analysis. I had never seen any topology before so the professor told me to work through the first two chapters of Hocking's Topology.

I am taking my sweet time with it because I want to be able to absorb everything. I was doing fine until recently when I got to basis and subbassis of a topology.

"Can we define a topology S for which each [itex]X_{a}[/itex] is an open set? The answer is yes because we may always assign to S the discrete topology in which there are no limit points. In the discrete topology every set is closed, hence every set is open."

My head exploded from reading that. Could someone please explain to me what the heck happened?