What is the proof for showing a space is connected?

[Zhalfirin88]

Homework Statement

Show that X is connected if and only if the only subsets of X that are both open and closed are the empty set and X.

Proof: https://files.nyu.edu/eo1/public/Book-PDF/Appendix.pdf
Page 14.

I'm confused by this proof. First, if S is not in {null set, X} then how can S be a subset of X?
Secondly, how can X = S U (X\S)?

 

[Office_Shredder]

Suppose that X is the set of real numbers. Then we could let S be the set [0,1]. {null,X} contains only two elements: the null set, and the set of real numbers, neither of which are [0,1]

 

[Zhalfirin88]

OK thanks. I didn't consider X to be a single element, then any nonempty subset of X would not be in {null, X}