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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

Relatively compact subset

A subset $A$ of a topological space $X$ with the property that the closure $\bar A$ of $A$ in $X$ is 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 $X$.

A space is compact if it is relatively compact in itself.

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 $X$.

References

• 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 MR0614752 Zbl 0452.06001

Core-compact space

MSC 54D30 54D50

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 in $V$ (every open cover of $V$ has a finite subset that covers $N$); equivalently, $N$ is way below $X$.

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.

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 ${-}^X$. See Exponential law (in topology).

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: Cambridge University Press (2014) (English) 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

Approach space

MSC 54A05 54E05

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 ${}^c$ denotes the Kuratowksi closure operator, via $$ d(x,A) = \begin{cases} 0 & \ \text{if}\ x \in A^c \\ \infty & \ \text{otherwise} \end{cases} \ . $$

In the opposite direction, if $(X,d)$ is an approach space then the operation $$ A^c = \{ x \in X : d(x,A) < \infty \} $$ is a Čech closure operator, giving $X$ the structure of a pre-topological space. However, the operation $$ A^C = \{ x \in X : d(x,A) = 0 \} $$ is a closure operator giving a topological structure on $X$.

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
• R. Lowen, "Approach spaces - a common supercategory of TOP and MET." Math. Nachr. 141 (1989) 183-226 Zbl 0676.54012
• R. Lowen, "Index Analysis: Approach Theory at Work", Springer (2015) ISBN 1-4471-6485-7 Zbl 1311.54002

Binary tetrahedral group

The exceptional group $G_4$ or $\langle 3,3,2 \rangle$, abstractly presented as: $$ \langle R,S \ |\ R^3=S^3=(RS)^2 \rangle \ . $$ 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 numbers: $$ \pm 1\,,\ \pm i\,,\ \pm j\,,\ \pm k\,,\ \frac{\pm1\pm i\pm j\pm k}{2} \ . $$

The group has an action on the three-sphere with tetrahedral space as quotient.

References

[a1]	H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1991) pp. 76 ISBN 0-521-20125-X Zbl 0732.51002

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

[a1]	H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1991) pp. 77 ISBN 0-521-20125-X Zbl 0732.51002

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

[a1]	H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1991) pp. 77 ISBN 0-521-20125-X Zbl 0732.51002

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

Dicyclic group

MSC 20F05

A finite group of order $4n$, obtained as the extension 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 quaternions.

The dicyclic group $n=2$ is the quaternion group of order $8$.

References

[a1]	H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1974) ISBN 0-521-20125-X Zbl 0732.51002

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

Compactly generated space

Kelley space, $k$-space

A Hausdorff topological space in which a subset is closed if its intersection with any compact subset is closed. Every locally compact Hausdorff space is compactly generated, as is every first countable Hausdorff space.

The category of compactly generated spaces and continuous maps is equivalent to the category of Hausdorff spaces and compactly continuous maps. It is a Cartesian-closed category.

See: Exponential law (in topology) and Space of mappings, topological.

References

• Francis Borceux, "Handbook of Categorical Algebra: Volume 2, Categories and Structures", Encyclopedia of Mathematics and its Applications, Cambridge University Press (1994) ISBN 0-521-44179-X Zbl 1143.18002

Compactly continuous map

A map $f$ of topological spaces $X \rightarrow Y$ with the property that the restriction of $f$ to any compact subspace of $X$ is continuous. Clearly any continuous map is compactly continuous, and the converse holds if $X$ is a locally compact space. The composite of compactly continuous maps is again compactly continuous.

The category of Hausdorff spaces and compactly continuous maps is equivalent to the category of compactly generated spaces and continuous maps. It is a Cartesian-closed category.

See: Exponential law (in topology) and Space of mappings, topological.

References

• Francis Borceux, "Handbook of Categorical Algebra: Volume 2, Categories and Structures", Encyclopedia of Mathematics and its Applications, Cambridge University Press (1994) ISBN 0-521-44179-X Zbl 1143.18002