mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real...

a dense order on the rational numbers. The real numbers form an initial unbounded totally ordered set that is connected in the order topology (defined...

In general topology, an Alexandrov topology is a topology in which the intersection of an arbitrary family of open sets is open (while the definition of...

set. Within the set of real numbers, either with the ordinary topology or the order topology, 0 is also a limit point of the set. It is also a limit point...

order is also important for identifying suitable topologies on partially ordered sets, as is done in order theory. Consider any topological space X. The...

≤). The finest order consistent topology is the Scott topology, which is coarser than the Alexandrov topology. A third important topology in this spirit...

Alexandrov topology Lexicographic order topology on the unit square Order topology Lawson topology Poset topology Upper topology Scott topology Scott continuity...

topological field. The Harrison topology is a topology on the set of orderings XF of a formally real field F. Each order can be regarded as a multiplicative...

In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly...

or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus,...

redirect targets Ordered vector space – Vector space with a partial order Poset topology, a kind of topological space that can be defined from any poset Scott...

(pre)order Order topology of a total order (open interval topology) Alexandrov topology Upper topology Scott topology Scott continuity Lawson topology Finer...

Lexicographic order topology on the unit square Right order topology Right order topology on R Right half-open interval topology Nested interval topology Overlapping...

general topology, a field of mathematics. Alexandrov topology Cantor space Co-kappa topology Cocountable topology Cofinite topology Compact-open topology Compactification...

Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric...

In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can...

In general topology, the lexicographic ordering on the unit square (sometimes the dictionary order on the unit square) is a topology on the unit square...

Lexicographic order topology on the unit square Lexicographic ordering in tensor abstract index notation Lexicographically minimal string rotation Leximin order Long...

elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy...

general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It...

) , {\displaystyle [0,1),} equipped with the order topology that arises from the lexicographical order on ω 1 × [ 0 , 1 ) {\displaystyle \omega _{1}\times...

mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning...

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean...

topology τ of a topological space (X, τ) is a family B {\displaystyle {\mathcal {B}}} of open subsets of X such that every open set of the topology is...

Convexity can be extended for a totally ordered set X endowed with the order topology. Let Y ⊆ X. The subspace Y is a convex set if for each pair of points...

James (2020), An Introduction to Order Theory, AMS Stone, M. H. (1934), "Boolean Algebras and Their Application to Topology", Proc. Natl. Acad. Sci. U.S.A...

reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion). In...

In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space ( X , ≤ ) {\displaystyle (X,\leq )}...

partial lattices: not every pair of elements has a meet or join. Pointless topology Lattice of subgroups Spectral space Invariant subspace Closure operator...

) {\displaystyle f(s)\leq g(s)} almost everywhere. Order topology (functional analysis) – Topology of an ordered vector space Ordered field – Algebraic...