Difference between revisions of "User:Richard Pinch/sandbox-10"
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| − | = | + | =Cantor–Bendixson characteristics= |
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| + | [[Ordinal number]] invariants of a [[Boolean algebra]]. | ||
| + | Let $B$ be a Boolean algebra, and $I(B)$ the [[ideal]] generated by the [[atom]]s. We have $I(B) = B$ if and only if $B$ is finite. We recursively define ideals $I_\alpha$ for [[ordinal number]]s $\alpha$, 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 $\beta$. | |
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| + | 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 $X$. | |
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| − | + | 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==== | ====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 space|scattered]]: has no dense-in-itself subset. | ||
| − | + | Countable superatomic Boolean algebras are determined up to isomorphism by their [[Cantor–Bendixson characteristics]]. | |
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| − | + | ====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. | |
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Latest revision as of 12:35, 18 January 2021
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.
Richard Pinch/sandbox-10. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-10&oldid=42505