User:Richard Pinch/sandbox-10

Cantor–Bendixson characteristics

Ordinal number invariants of a Boolean algebra. Let $B$ be a Boolean algebra, and $I(B)$ the ideal generated by the atoms. We have $I(B) = B$ if and only if $B$ is finite. We recursively define ideals $I_\alpha$ for ordinal numbers $\alpha$, together with homomorphisms $\pi_\alpha$ and algebras $B_\alpha$ with $\pi_\alpha : B \rightarrow B_\alpha$ with kernel $I_\alpha$, as follows: $I_0(B) =\{0\}$; if $\alpha = \beta+1$ then $I_\alpha = \pi_\beta^{-1}(B_\beta)$ and if $\alpha$ is a limit ordinal then $I_\alpha = \cup_{\beta<\alpha} I_\beta$. There is a least $\alpha$ such that $I_\alpha = I_\gamma$ for all $\gamma > \alpha$.

If $B$ is a superatomic Boolean algebra then each $A_\alpha$ is atomic and the sequence $I_\alpha$ stabilises at $\alpha$ with $\alpha=\beta+1$ where $\beta$ is the least ordinal such that $A_\beta$ is finite. The Cantor–Bendixson characteristics of $A$ are the quantities $(\beta,n)$ where $n$ is the number of atoms in $A_\beta$. The Cantor–Bendixson height or rank is $\beta$.

For countable superatomic Boolean algebras, the Cantor–Bendixson characteristics determine the isomorphism class.

For a topological space $X$, we may analogously define a Cantor–Bendixson rank as follows. Let $X_0 = X$ and for ordinal $alpha$ let $X_{\alpha+1}$ be the derived set of $X_\alpha$. If $\lambda$ is a limit ordinal, let $X_\lambda = \cap_{\alpha<\lambda} X_\alpha$. The sequence $(X_\alpha)$ is descending and the smallest $\alpha$ such that $X_{\alpha+1} = X_\alpha$ is the Cantor–Bendixson rank of $X$.

An analogous definition can be made for any partial ordered set $(A,{<})$ equipped with a map $f:A\rightarrow A$ such that $f(x) \le x$.

References

• Winfried Just, Martin Weese, "Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician", American Mathematical Society (1997) ISBN 0-8218-7208-7 Zbl 0887.03036

Superatomic Boolean algebra

A Boolean algebra for which every homomorphic image is atomic. Equivalently, the Stone space is scattered: has no dense-in-itself subset.

Countable superatomic Boolean algebras are determined up to isomorphism by their Cantor–Bendixson characteristics.

References

• Winfried Just, Martin Weese, "Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician", American Mathematical Society (1997) ISBN 0-8218-7208-7 Zbl 0887.03036
• J. Roitman, "Superatomic Boolean algebras" J.D. Monk (ed.) R. Bonnet (ed.) , Handbook of Boolean algebras , 1–3 , North-Holland (1989) pp. Chapt. 19; pp. 719–740 Zbl 0671.06001

S-space

A topological space which is regular Hausdorff hereditarily separable but not hereditarily Lindelöf. Dually, an L-space is regular Hausdorff hereditarily Lindelöf but not hereditarily separable. The question of the existence of S-spaces and L-spaces is connected to the Suslin problem. A Suslin line is an L-space, and an S-space may be constructed from a Suslin line. It is know that non-existence of an S-space is consistent with ZFC.