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=Core-compact space=
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=Way below=
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Let $X$ be a topological space with $\mathfrak{O}_X$ the collection of open sets.  If $U, V$ are open, we say that $U$ is compact in $V$ if every open cover of $V$ has a finite subset that covers $U$.  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 compact in $V$. 
 
  
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A space is core compact if and only if $\mathfrak{O}_X$ 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.
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MSC 06A06 06B35
  
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A space is core compact if and only if the product of the identity with a quotient map is quotient.
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''essentially below''
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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 $a$.
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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 $x$.
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====References====
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* 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}}
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* 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}}
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=Downset=
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MSC 06A06
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''lower set'', ''lower cone''
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A subset $S$ of a [[partially ordered set]] $(P,{\le})$ with the property that if $x \in S$ and $y \le x$ then $y \in S$.
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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, $A = A^\Delta$.
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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 $x^\nabla = \{y \in P : x \le y \}$.
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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]].
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==References==
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* 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}}
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* 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}}
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=Developable space=
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A '''development''' in a [[topological space]] $X$ is a sequence of [[open cover]]s $G_n$ such that for all points $x \in X$ the stars
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$$
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\mathrm{St}(x,G_n) = \cup \{ U \in G_n : x \in U \}
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$$
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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 space|collection-wise normal]] Moore space is [[Metrizable space|metrizable]].
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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.
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====References====
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* 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. [http://gallica.bnf.fr/ark:/12148/bpt6k3130n.f1451] {{ZBL|49.0702.06}}  {{ZBL|50.0696.01}}
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* Bing, R.H.  "Metrization of topological spaces", ''Canad. J. Math.'' '''3''' (1951) 175-186 {{DOI|10.4153/CJM-1951-022-3}} {{ZBL|0042.41301}}
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=Scott topology=
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MSC 06F30
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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 $F \subseteq S$.
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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.
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==References==
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* 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}}
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* 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}}

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

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
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