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What about fractals? Or more general, objects with some kind of self similarity? They often arise as fixed points of a self-map. In category theory these are known as initial algebras or terminal coalgebras. For example, $\mathbb{N}=1+\mathbb{N}$, $[0,1] = [0,1] {\cup}_{0 \sim 1} [0,1]$, and the set of binary trees $T$ satisfies $T=1+T^2$. Another example which comes to my mind: There are abelian groups $A$ with $A^3 \cong A$ and $A^2 \not\cong A$. I have to admit that this does not quite fit to your question, since you want that "$X \\in X$", but in the above examples we have "$X \subsetneq X$". So let me add something else:

• The category of small categories, functors and morphism of functors $\mathsf{Cat}$ is a category (which is not small).
• The class of ordinal numbers $\mathrm{On}$ is with $\in$ the well-order of all isomorphism classes of small well-orders. Of course it is not small.
• Quite similar and trivial, but the von Neumann universe is just the set (or class if you don't work with universes) of all small sets.
• If $X$ is a set, the set of topologies on $X$ carries a topology. A subbase is given by those topologies containing some fixed subset of $X$.
• Uniform convergence spaces seem to be filters of filters(?).
• This paper discusses the graph of all graphs on $n$ vertices on page 8.
• The Rado graph contains a copy of every finite or countably infinite graph.

What about fractals? Or more general, objects with some kind of self similarity? They often arise as fixed points of a self-map. In category theory these are known as initial algebras or terminal coalgebras. For example, $\mathbb{N}=1+\mathbb{N}$, $[0,1] = [0,1] {\cup}_{0 \sim 1} [0,1]$, and the set of binary trees $T$ satisfies $T=1+T^2$. Another example which comes to my mind: There are abelian groups $A$ with $A^3 \cong A$ and $A^2 \not\cong A$. I have to admit that this does not quite fit to your question, since you want that "$X \\in X$", but in the above examples we have "$X \subsetneq X$". So let me add something else:

• The category of small categories, functors and morphism of functors $\mathsf{Cat}$ is a category (which is not small).
• The class of ordinal numbers $\mathrm{On}$ is with $\in$ the well-order of all isomorphism classes of small well-orders. Of course it is not small.
• Quite similar and trivial, but the von Neumann universe is just the set (or class if you don't work with universes) of all small sets.
• If $X$ is a set, the set of topologies on $X$ carries a topology. A subbase is given by those topologies containing some fixed subset of $X$.
• Uniform convergence spaces seem to be filters of filters(?).
• This paper discusses the graph of all graphs on $n$ vertices on page 8.
• The Rado graph contains a copy of every finite or countably infinite graph.

What about fractals? Or more general, objects with some kind of self similarity? They often arise as fixed points of a self-map. In category theory these are known as initial algebras or terminal coalgebras. For example, $\mathbb{N}=1+\mathbb{N}$, $[0,1] = [0,1] {\cup}_{0 \sim 1} [0,1]$, and the set of binary trees $T$ satisfies $T=1+T^2$. Another example which comes to my mind: There are abelian groups $A$ with $A^3 \cong A$ and $A^2 \not\cong A$. I have to admit that this does not quite fit to your question, since you want that "$X \\in X$", but in the above examples we have "$X \subsetneq X$". So let me add something else:

• The category of small categories, functors and morphism of functors $\mathsf{Cat}$ is a category (which is not small).
• The class of ordinal numbers $\mathrm{On}$ is with $\in$ the well-order of all isomorphism classes of small well-orders. Of course it is not small.
• Quite similar and trivial, but the von Neumann universe is just the set (or class if you don't work with universes) of all small sets.
• If $X$ is a set, the set of topologies on $X$ carries a topology. A subbase is given by those topologies containing some fixed subset of $X$.
• Uniform convergence spaces seem to be filters of filters(?).

What about fractals? Or more general, objects with some kind of self similarity? They often arise as fixed points of a self-map. In category theory these are known as initial algebras or terminal coalgebras. For example, $\mathbb{N}=1+\mathbb{N}$, $[0,1] = [0,1] {\cup}_{0 \sim 1} [0,1]$, and the set of binary trees $T$ satisfies $T=1+T^2$. Another example which comes to my mind: There are abelian groups $A$ with $A^3 \cong A$ and $A^2 \not\cong A$. I have to admit that this does not quite fit to your question, since you want that "$X \\in X$", but in the above examples we have "$X \subsetneq X$". So let me add something else:

• The category of small categories, functors and morphism of functors $\mathsf{Cat}$ is a category (which is not small).
• The class of ordinal numbers $\mathrm{On}$ is with $\in$ the well-order of all isomorphism classes of small well-orders. Of course it is not small.
• Quite similar and trivial, but the von Neumann universe is just the set (or class if you don't work with universes) of all small sets.
• If $X$ is a set, the set of topologies on $X$ carries a topology. A subbase is given by those topologies containing some fixed subset of $X$.
• Uniform convergence spaces seem to be filters of filters(?).
• This paper discusses the graph of all graphs on $n$ vertices on page 8.
• The Rado graph contains a copy of every finite or countably infinite graph.

What about fractals? Or more general, objects with some kind of self similarity? They often arise as fixed points of a self-map. In category theory these are known as initial algebras or terminal coalgebras. For example, $\mathbb{N}=1+\mathbb{N}$, $[0,1] = [0,1] {\cup}_{0 \sim 1} [0,1]$, and the set of binary trees $T$ satisfies $T=1+T^2$. Another example which comes to my mind: There are abelian groups $A$ with $A^3 \cong A$ and $A^2 \not\cong A$. I have to admit that this does not quite fit to your question, since you want that "$X \\in X$", but in the above examples we have "$X \subsetneq X$". So let me add something else:

• The category of small categories, functors and morphism of functors $\mathsf{Cat}$ is a category (which is not small).
• The class of ordinal numbers $\mathrm{On}$ is with $\in$ the well-order of all isomorphism classes of small well-orders. Of course it is not small.
• Quite similar and trivial, but the von Neumann universe is just the set (or class if you don't work with universes) of all small sets.
• If $X$ is a set, the set of topologies on $X$ is again a topology. A subbase is given by those topologies containing some fixed subset of $X$.
• Uniform convergence spaces seem to be filters of filters(?).

What about fractals? Or more general, objects with some kind of self similarity? They often arise as fixed points of a self-map. In category theory these are known as initial algebras or terminal coalgebras. For example, $\mathbb{N}=1+\mathbb{N}$, $[0,1] = [0,1] {\cup}_{0 \sim 1} [0,1]$, and the set of binary trees $T$ satisfies $T=1+T^2$. Another example which comes to my mind: There are abelian groups $A$ with $A^3 \cong A$ and $A^2 \not\cong A$. I have to admit that this does not quite fit to your question, since you want that "$X \\in X$", but in the above examples we have "$X \subsetneq X$". So let me add something else:

• The category of small categories, functors and morphism of functors $\mathsf{Cat}$ is a category (which is not small).
• The class of ordinal numbers $\mathrm{On}$ is with $\in$ the well-order of all isomorphism classes of small well-orders. Of course it is not small.
• Quite similar and trivial, but the von Neumann universe is just the set (or class if you don't work with universes) of all small sets.
• If $X$ is a set, the set of topologies on $X$ carries a topology. A subbase is given by those topologies containing some fixed subset of $X$.
• Uniform convergence spaces seem to be filters of filters(?).