In the context of Dynkin system D, is D \ $\emptyset$ the universal set $\Omega$? A Dynkin system, is a collection of subsets of another universal set ${\displaystyle \Omega }$ satisfying a set of axioms weaker than those of σ-algebra. here is one of the Definitions that wiki gives > Let Ω be a nonempty set, and let ${\displaystyle D}$ be a collection of subsets of Ω. Then D is a Dynkin system if > > 1. Ω ∈ D > > 2. if A ∈ D, then $A^c$ ∈ D > > 3. if $A_1, A_2, A_3, ...$ is a sequence of subsets in D such that $A_i ∩ A_j$ = Ø, for all i ≠ j, then ${\displaystyle \bigcup _{n=1}^{\infty }A_{n}\in D} $ > > The book "Infinite Dimensional Analysis: A Hitchhiker's Guide" gives a set > X = {1,2,3,4}. and claims > A = {$\emptyset$, {1,2},{3,4},{1,3},{2,4},X} is a Dynkin system that is neigher an algebra nor a π-system in this case, is the universal set $\Omega$ {{1,2},{3,4},{1,3},{2,4},X}, namely, is D \ $\emptyset$ the universal set $\Omega$?

Here the universal set is $X = \\{1,2,3,4\\}$

So the Dynkin system, as the definition, is a collection of subsets of $X$.

$A$ is a Dynkin system by checking the definition. (1) it contains the universal set $X$ (2) it is closed under complement operation (3) union of mutual disjoint elements in $A$ is still in $A$.

$A$ is not a $\pi$-system since it is not closed under finite intersection: $\\{1,2\\}\cap\\{2,4\\} = \\{2\\}\
otin A$

$A$ is not an algebra since it is not closed under finite union: $\\{1,2\\}\cup\\{2,4\\} = \\{1,2,4\\}\
otin A$