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=Cantor–Bendixson characteristics= | =Cantor–Bendixson characteristics= | ||
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 $\gamma > \alpha$. | 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 $\gamma > \alpha$. | ||
Revision as of 19:50, 17 January 2021
Cantor–Bendixson characteristics
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 is $\beta$.
For countable superatomic Boolean algebras, the Cantor–Bendixson characteristics determine the isomorphism class.
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
Separated space
A separated space may refer to
- a topological space satisfying a separation axiom; in particular a Hausdorff space;
- a left separated space or right separated space.
Right separated space
A topological space $X$ is right (resp. left) separated if there is a well ordering ${<}$ on $X$ such that the segments $\{x \in X : x < y\}$ are all open (resp. closed) in the topology of $X$.
A Hausdorff space is scattered if and only if it is right separated.
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=51380