In the following lecture notes one finds the following:
Examples. The following are all examples of σ-algebras. (HW: check this in each case) • Let P(X) denote the collection of all subsets of X, i.e. P(X) := {A : A ⊂ X} (we write ⊂ rather than ⊆, so in our notation X ⊂ X is a true statement). P(X) is called the power set of X, and is a σ-algebra.
I don't understand how using the strict subset makes X ⊂ X a true statement, or what the author is trying to get across here?
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$\begingroup$Note that the author says:
"we write $\subset$ rather than $\subseteq$, so in our notation $X\subset X$ is a true statement."
They are using the symbol "$\subset$" as a synonym for "$\subseteq$." (This leaves the symbol "$\subsetneq$" for proper subsethood.) Munrkes' topology textbook also follows this convention.
Personally I think this is a terrible choice since it clashes with "$<$ vs. $\le$," but they are being consistent and stating it explicitly.
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