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=Nephroid=
 
An [[epicycloid]] with parameter $m=2$; an algebraic plane curve with equation
 
$$
 
x= 3r \cos\theta-r\cos\left[3\theta\right] \,,
 
$$
 
$$
 
y= 3r \sin\theta-r\sin\left[3\theta\right] \ .
 
$$
 
 
The nephroid is the [[catacaustic]] of the [[cardioid]] with respect to a cusp, and of a circle with respect to a point at infinity; the [[evolute]] of a nephroid is another nephroid.
 
 
The '''nephroid of Freeth''' is the [[strophoid]] of a circle with respect to its centre and a point on the circumference.  It has equation
 
$$
 
r = a(1 + 2\sin(\theta/2)) \ .
 
$$
 
 
====References====
 
* J.D. Lawrence,  "A catalog of special plane curves" , Dover  (1972) ISBN 0-486-60288-5  {{ZBL|0257.50002}}
 
 
 
 
=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

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.

How to Cite This Entry:
Richard Pinch/sandbox-10. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-10&oldid=51380