Latest revision as of 19:25, 20 January 2021

Way below

MSC 06A06 06B35

essentially below

Let $(X,{\le})$ be a partially ordered set. The way below relationship $\ll$ determined by ${\le}$ is defined as $x \ll y$ if for each up-directed subset $D$ of $X$ for which $y \le \sup D$, there is a $d \in D$ such that $x \le d$. Write $\Downarrow y = \{ x : x \ll y \}$: this is an ideal, indeed, the intersection of all ideals $I$ with $y \le \sup I$. A continuous lattice is one in which $a = \sup \Downarrow a$ for all $a$.

A compact element $x \in X$ is one for which $x \ll x$. An ordered set is complete if $x = \sup\Downarrow x$ for all $x$.

References

G. Gierz, Karl Heinrich Hofmann, K. Keimel, J.D. Lawson, M. Mislove, Dana S. Scott, "A compendium of continuous lattices" Springer (1980) ISBN 3-540-10111-X MR0614752 Zbl 0452.06001
Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology." Encyclopedia of Mathematics and its Applications 153 Cambridge (2014) ISBN 978-1-107-06394-5 Zbl 1297.18001

Downset

MSC 06A06

lower set, lower cone

A subset $S$ of a partially ordered set $(P,{\le})$ with the property that if $x \in S$ and $y \le x$ then $y \in S$.

The principal downset on an element $a \in P$ is the set $x^\Delta$, also denoted $(x]$, defined as $x^\Delta = \{y \in P : y \le x \}$. The down-closure of a set $A$ is $A^\Delta = \cup_{x \in A}\, x^\Delta$. A set $A$ is a downset if and only if it is equal to its down-closure, $A = A^\Delta$.

The dual notion of upset (upper set, upper cone) is defined as a subset $S$ of with the property that if $x \in S$ and $x \le y$ then $y \in S$. The principal upset on an element $a \in P$ is the set $x^\nabla$, also denoted $[x)$, defined as $x^\nabla = \{y \in P : x \le y \}$.

The terms "ideal" and "filter" are sometimes used for downset and upset respectively. However, it is usual to impose the extra condition that an ideal contain the supremum of any two elements (or up directed) and, dually, that a filter contain the infimum of any two element (or down directed). See the comments at Ideal and Filter.

References

B. A. Davey, H. A. Priestley, Introduction to lattices and order, 2nd ed. Cambridge University Press (2002) ISBN 978-0-521-78451-1 Zbl 1002.06001
Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology." Encyclopedia of Mathematics and its Applications 153 Cambridge (2014) ISBN 978-1-107-06394-5 Zbl 1297.18001

Developable space

A development in a topological space $X$ is a sequence of open covers $G_n$ such that for all points $x \in X$ the stars $$ \mathrm{St}(x,G_n) = \cup \{ U \in G_n : x \in U \} $$ form a local base for $x$. A developable space is a space with a development. A metric space is a developable space: the sequence of collections of open balls of radius $1/n$ forming a development. A Moore space is a regular space with a development. A collection-wise normal Moore space is metrizable.

A regular development has the further property that if $U,V \in G_{n+1}$ with $U \cap V \neq \emptyset$, then there is $W \in G_n$ with $U \cup V \subset W$. Alexandroff and Urysohn proved that a space is metrizable if and only if it has a regular development.

References

Alexandroff, P.; Urysohn, P. "Une condition nécessaire et suffisante pour qu’une classe $(\mathcal{L})$ doit une classe $(\mathcal{B})$", Comptes Rendus 177 (1923) 1274-1276. [1] Zbl 49.0702.06 Zbl 50.0696.01
Bing, R.H. "Metrization of topological spaces", Canad. J. Math. 3 (1951) 175-186 DOI 10.4153/CJM-1951-022-3 Zbl 0042.41301

Scott topology

MSC 06F30

A topology on a partially ordered set $(X,{\le})$ for which the open sets are the Scott open subsets: a downset $U$ is Scott open if for any set $S$ of $X$ with $\wedge S \in U$ then $\wedge F \in U$ for some finite $F \subseteq S$.

A function between partially ordered sets is Scott continuous in the Scott topologies if and only if it preserves meets of down-directed sets.

References

G. Gierz, Karl Heinrich Hofmann, K. Keimel, J.D. Lawson, M. Mislove, Dana S. Scott, "A compendium of continuous lattices" Springer (1980) ISBN 3-540-10111-X MR0614752 Zbl 0452.06001
Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology." Encyclopedia of Mathematics and its Applications 153 Cambridge (2014) ISBN 978-1-107-06394-5 Zbl 1297.18001

How to Cite This Entry:
Richard Pinch/sandbox-9. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-9&oldid=42461

@@ Line 1: / Line 1: @@
 =Way below=
-Let $(X,{\le})$ be a [[partially ordered set]].  The way below relationship $\ll$ determined by ${\le}$ is defined as $x \ll y$ if for each up directed subset $D$ of $X$ for which $y \le \sup D$, there is a $d \in D$ such that $x \le d$.  Write $\Downarrow y = \{ x : x \ll y \}$.
-A ''compact'' element $x \in X$ is one for which $x \ll x$.  An ordered set is complete if $x = \sup\Downarrow x$ for all $x$.
+MSC 06A06 06B35
+''essentially below''
+Let $(X,{\le})$ be a [[partially ordered set]].  The way below relationship $\ll$ determined by ${\le}$ is defined as $x \ll y$ if for each up-[[Directed set|directed subset]] $D$ of $X$ for which $y \le \sup D$, there is a $d \in D$ such that $x \le d$.  Write $\Downarrow y = \{ x : x \ll y \}$: this is an [[ideal]], indeed, the intersection of all ideals $I$ with $y \le \sup I$.  A ''[[continuous lattice]]'' is one in which $a = \sup \Downarrow a$ for all $a$.
+A ''compact'' element $x \in X$ is one for which $x \ll x$.  An ordered set is ''complete'' if $x = \sup\Downarrow x$ for all $x$.
 ====References====
 * G. Gierz, Karl Heinrich Hofmann, K. Keimel, J.D. Lawson, M. Mislove, Dana S. Scott, "A compendium of continuous lattices" Springer (1980)  ISBN 3-540-10111-X  {{MR|0614752}}  {{ZBL|0452.06001}}
-* Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology."  Encyclopedia of Mathematics and its Applications '''153''' Cambridge: Cambridge University Press (2014) (English) ISBN 978-1-107-06394-5 {{ZBL|1297.18001}}
+* Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology."  Encyclopedia of Mathematics and its Applications '''153''' Cambridge (2014) ISBN 978-1-107-06394-5 {{ZBL|1297.18001}}
-=Relatively compact subset=
+=Downset=
-A subset $A$ of a [[topological space]] $X$ with the property that the [[Closure of a set|closure]] $\bar A$ of $A$ in $X$ is [[Compact space|compact]].
-A subset $A$ of a metric space $X$ is relatively compact if and only if every sequence of points in $A$ has a cluster point in $X$.
+MSC 06A06
-A space is compact if it is relatively compact in itself.
+''lower set'', ''lower cone''
-An alternative definition is that $A$ is relatively compact in $X$ if and only if every open cover of $X$ contains a finite subcover of $A$.  This formulation is equivalent to requiring that the set $A$ be [[way below]] $X$ with respect to set inclusion and the directed set of open subsets of $X$.
+A subset $S$ of a [[partially ordered set]] $(P,{\le})$ with the property that if $x \in S$ and $y \le x$ then $y \in S$.
-====References====
+The principal downset on an element $a \in P$ is the set $x^\Delta$, also denoted $(x]$, defined as $x^\Delta = \{y \in P : y \le x \}$.  The down-closure of a set $A$ is $A^\Delta = \cup_{x \in A}\, x^\Delta$.  A set $A$ is a downset if and only if it is equal to its down-closure, $A = A^\Delta$.
-* N. Bourbaki, "General Topology" Volume 4 Ch.5-10, Springer [1974] (2007) ISBN 3-540-34399-7 {{ZBL|1107.54002}}
-* G. Gierz, Karl Heinrich Hofmann, K. Keimel, J.D. Lawson, M. Mislove, Dana S. Scott, "A compendium of continuous lattices" Springer (1980)  ISBN 3-540-10111-X  {{MR|0614752}}  {{ZBL|0452.06001}}
-=Core-compact space=
+The dual notion of ''upset'' (''upper set'', ''upper cone'') is defined as a subset $S$ of with the property that if $x \in S$ and $x \le y$ then $y \in S$.  The principal upset on an element $a \in P$ is the set $x^\nabla$, also denoted $[x)$, defined as $x^\nabla = \{y \in P : x \le y \}$.
-Let $X$ be a topological space.  The space $X$ is core compact if for any $x \in X$ and open neighbourhood $N$ of $x$, there is an open set $V$ such that $N$ is [[Relatively-compact set|relatively compact]] in $V$ (every open cover of $V$ has a finite subset that covers $N$).
-A space is core compact if and only if the collection of open sets $\mathfrak{O}_X$ is a [[continuous lattice]].  A [[locally compact space]] is core compact, and a [[sober space]] (and hence in particular a [[Hausdorff space]]) is core compact if and only if it is locally compact.
+The terms "ideal" and "filter" are sometimes used for downset and upset respectively.  However, it is usual to impose the extra condition that an ideal contain the supremum of any two elements (or up [[Directed order|directed]]) and, dually, that a filter contain the infimum of any two element (or down directed).  See the comments at [[Ideal]] and [[Filter]].
-A space is core compact if and only if the product of the identity with a quotient map is quotient.  The core compact spaces are precisely the exponentiable spaces in the [[category]] of topological spaces; that is, the spaces $X$ such that ${-} \times X$ has a right adjoint ${-}^X$.
+==References==
+* B. A. Davey, H. A. Priestley, ''Introduction to lattices and order'', 2nd ed. Cambridge University Press  (2002) ISBN 978-0-521-78451-1 {{ZBL|1002.06001}}
-====References====
+* Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology."  Encyclopedia of Mathematics and its Applications '''153''' Cambridge (2014) ISBN 978-1-107-06394-5 {{ZBL|1297.18001}}
-* Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology."  Encyclopedia of Mathematics and its Applications '''153''' Cambridge: Cambridge University Press (2014) (English) ISBN 978-1-107-06394-5 {{ZBL|1297.18001}}
 =Developable space=
@@ Line 44: / Line 44: @@
 * Bing, R.H.  "Metrization of topological spaces", ''Canad. J. Math.'' '''3''' (1951) 175-186 {{DOI|10.4153/CJM-1951-022-3}} {{ZBL|0042.41301}}
-=Approach space=
+=Scott topology=
-A generalisation of the concept of [[metric space]], formalising the notion of the distance from a point to a set.  An approach space is a set $X$ together with a function $d$ on $X \times \mathcal{P}X$, where $\mathcal{P}X$ is the [[power set]] of $X$, talking values in the extended positive reals $[0,\infty]$, and satisfying
-$$
-d(x,\{x\}) = 0 \ ;
-$$
-$$
-d(x,\emptyset) = \infty \ ;
-$$
-$$
-d(x,A\cup B) = \min(d(x,A),d(x,B)) \ ;
-$$
-$$
-d(x,A) \le d(x,A^u) + u \ ;
-$$
-where for $u \in [0,\infty]$, we write $A^u = \{x \in X : d(x,A) \le u \}$.
-A metric space $(X,\delta)$ has an approach structure via
-$$
-d(x,A) = \inf\{ \delta(x,a) : a \in A \} \ .
-$$
-and a topological space $X,{}^c$, where ${}^c$ is the [[Kuratowksi closure operator]], via
-$$
-d(x,A) = \begin{cases} 0 & \ \text{if} x \in A^c \\ \infty & \ \text{otherwise} \end{cases} \ .
-$$
-====References====
-* Hofmann, Dirk (ed.); Seal, Gavin J. (ed.); Tholen, Walter (ed.) "Monoidal topology. A categorical approach to order, metric, and topology" Cambridge University Press (2014) ISBN 978-1-107-06394-5 {{ZBL|1297.18001}}
+MSC 06F30
-=Ample field=
+A topology on a partially ordered set $(X,{\le})$ for which the open sets are the ''Scott open'' subsets: a [[downset]] $U$ is Scott open if for any set $S$ of $X$ with $\wedge S \in U$ then $\wedge F \in U$ for some finite $F \subseteq S$.
-A field which is [[existentially closed]] in its field of [[formal power series]].  Examples include [[pseudo algebraically closed field]]s, [[real closed field]]s and [[Henselian]] fields.
-A field $K$ is ample if and only if every absolutely irreducible curve over $K$ with a simple $K$-point has infinitely many $K$-points.
+A function between partially ordered sets is Scott continuous in the Scott topologies if and only if it preserves meets of down-[[directed set]]s.
-====References====
-* Moshe Jarden, "Algebraic patching", Springer (2011) ISBN 978-3-642-15127-9 {{ZBL|1235.12002}}
-=Binary tetrahedral group=
+==References==
-The exceptional group $G_4$ or $\langle 3,3,2 \rangle$, abstractly presented as:
+* G. Gierz, Karl Heinrich Hofmann, K. Keimel, J.D. Lawson, M. Mislove, Dana S. Scott, "A compendium of continuous lattices" Springer (1980)  ISBN 3-540-10111-X  {{MR|0614752}}  {{ZBL|0452.06001}}
-$$
+* Dirk Hofmann, Gavin J. Seal, Walter Thole (edd.) "Monoidal topology. A categorical approach to order, metric, and topology."  Encyclopedia of Mathematics and its Applications '''153''' Cambridge (2014) ISBN 978-1-107-06394-5 {{ZBL|1297.18001}}
-\langle R,S \ |\ R^3=S^3=(RS)^2 \rangle \ .
-$$
-It is finite of order 24.  It has the [[alternating group]] $A_4$ as quotient by the centre  and the [[quaternion group]] of order 8 as a quotient.
-This group may be realised as the group of invertible [[Hurwitz number]]s:
-$$
-\pm 1\,,\ \pm i\,,\ \pm j\,,\ \pm k\,,\ \frac{\pm1\pm i\pm j\pm k}{2} \ .
-$$
-The group has an action on the three-sphere with [[tetrahedral space]] as quotient.
-====References====
-<table>
-<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Regular complex polytopes" , Cambridge Univ. Press  (1991)  pp. 76 ISBN 0-521-20125-X  {{ZBL|0732.51002}}</TD></TR>
-</table>
-=Binary icosahedral group=
-The group $\langle 5,3,2 \rangle$ abstractly presented as:
-$$
-\langle A,B \ |\ A^5=B^3=(AB)^2 \rangle \ .
-$$
-It is finite of order 120.
-The group has an action on the three-sphere with [[dodecahedral space]] as quotient.
-====References====
-<table>
-<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Regular complex polytopes" , Cambridge Univ. Press  (1991)  pp. 77 ISBN 0-521-20125-X  {{ZBL|0732.51002}}</TD></TR>
-</table>
-=Binary octahedral group=
-The group $\langle 4,3,2 \rangle$ abstractly presented as:
-$$
-\langle A,B \ |\ A^4=B^3=(AB)^2 \rangle \ .
-$$
-It is finite of order 48.  It has the [[binary tetrahedral group]] $G_4 = \langle 3,3,2 \rangle$ as a subgroup of index 2.
-The group has an action on the three-sphere with [[octahedral space]] as quotient.
-====References====
-<table>
-<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Regular complex polytopes" , Cambridge Univ. Press  (1991)  pp. 77 ISBN 0-521-20125-X  {{ZBL|0732.51002}}</TD></TR>
-</table>
-=Dodecahedral space=
-The result of identifying opposite faces of a [[dodecahedron]] by a right-handed turn of angle $\pi/5$.  It is the quotient of the three-sphere by the [[binary icosahedral group]].
-Dodecahedral space is a homology sphere (Poincaré sphere).
-====References====
-* José Maria Montesinos, "Classical tessellations and three-manifolds" Springer (1987)  ISBN 3-540-15291-1 {{ZBL|0626.57002}}
-=Étale algebra=
-A commutative algebra $A$ finite-dimensional over a field $K$ for which the bilinear form induced by the trace
-$$
-\langle x,y \rangle = \mathrm{tr}_{A/K} (x\cdot y)
-$$
-is non-singular.  Equivalently, an algebra which is isomorphic to a product of field $A \sim K_1 \times \cdots \times K_r$ with each $K_i$ an extension of $K$.
-Since $\langle xy,z \rangle = \mathrm{tr}(xyz) = \langle x,yz \rangle$, an étale algebra is a [[Frobenius algebra]] over $K$.
-====References====
-* Tsit-Yuen Lam, "Lectures on Modules and Rings" Graduate Texts in Mathematics '''189''' Springer (2012) ISBN 1461205255 {{ZBL|0911.16001}}
-=Unit quaternion=
-A [[quaternion]] with norm 1, that is, $x_i + yj + zk + t$ with $x^2+y^2+z^2+t^2 = 1$.
-The real unit quaternions form a group isomorphic to the special unitary group $\mathrm{SU}_2$ over the complex numbers, and to the [[spin group]] $\mathrm{Sp}_3$.  They double cover the [[rotation group]] $\mathrm{SO}_3$ with kernel $\pm 1$.
-The finite subgroups of the unit quaternions are given by [[group presentation]]s
-$$
-A^p = B^q = (AB)^2
-$$
-with $1/p + 1/q > 1/2$, denoted $\langle p,q,2 \rangle$.  They are
-* the [[cyclic group]]s $C_n$, , corresponding to $\langle n,n,1 \rangle$;
-* the [[dicyclic group]]s, corresponding to $\langle n,2,2 \rangle$;
-* the [[binary tetrahedral group]] $\langle 3,3,2 \rangle$;
-* the [[binary octahedral group]] $\langle 4,3,2 \rangle$;
-* the [[binary icosahedral group]] $\langle 5,3,2 \rangle$.
-====References====
-<table>
-<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Regular complex polytopes" , Cambridge Univ. Press  (1991)  {{ZBL|0732.51002}}</TD></TR>
-</table>
-=Dicyclic group=
-A finite group of order $4n$, obtained as the extensions of the cyclic group of order $2$ by a cyclic group of order $2n$.  It has the presentation $\langle n,2,2 \rangle$ and [[group presentation]]
-$$
-A^n = B^2 = (AB)^2 \ .
-$$
-It may be realised as a subgroup of the [[unit quaternion]]s.
-The dicyclic group $n=2$ is the [[quaternion group]] of order $8$.
-====References====
-<table>
-<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Regular complex polytopes" , Cambridge Univ. Press  (1991)  {{ZBL|0732.51002}}</TD></TR>
-</table>

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Difference between revisions of "User:Richard Pinch/sandbox-9"

Latest revision as of 19:25, 20 January 2021

Contents

Way below

References

Downset

References

Developable space

References

Scott topology

References