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...

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...

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...

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,...

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

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

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

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

partial orderPages displaying short descriptions of redirect targets Ordered vector space – Vector space with a partial order Poset topology, a kind of...

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

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...

pointless topology, also called point-free topology (or pointfree topology) or topology without points and locale theory, is an approach to topology that avoids...

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

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

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...

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...

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

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...

induced order topology, i.e. the order topology of the totally ordered Y, where this order is inherited from X, is coarser than the subspace topology of the...

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

be given a topology, the Alexandrov topology; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set...

{\mathbb {R} }}} is given the left order topology. This is just a restatement of condition (2) since the left order topology is generated by all the intervals...

with the order topology; this topology is discrete if and only if it is less than or equal to ω. A subset of ω + 1 is open in the order topology if and...