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Though the subspace topology of Y = {−1} ∪ {1/n } n∈N in the section above is shown not to be generated by the induced order on Y, it is nonetheless an order topology on Y; indeed, in the subspace topology every point is isolated (i.e., singleton {y} is open in Y for every y in Y), so the subspace topology is the discrete topology on Y (the topology in which every subset of Y is open ...
In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space (,) is the finest locally convex topological vector space (TVS) topology on for which every order interval is bounded, where an order interval in is a set of the form [,]:= {:} where and belong to . [1]
The finest order consistent topology is the Scott topology, which is coarser than the Alexandrov topology. A third important topology in this spirit is the Lawson topology . There are close connections between these topologies and the concepts of order theory.
The canonical application of topological sorting is in scheduling a sequence of jobs or tasks based on their dependencies.The jobs are represented by vertices, and there is an edge from x to y if job x must be completed before job y can be started (for example, when washing clothes, the washing machine must finish before we put the clothes in the dryer).
See Scott topology. Scott topology. For a poset P, a subset O is Scott-open if it is an upper set and all directed sets D that have a supremum in O have non-empty intersection with O. The set of all Scott-open sets forms a topology, the Scott topology. Semilattice. A semilattice is a poset in which either all finite non-empty joins (suprema) or ...
The Harrison topology is a topology on the set of orderings X F of a formally real field F. Each order can be regarded as a multiplicative group homomorphism from F ∗ onto ±1. Giving ±1 the discrete topology and ±1 F the product topology induces the subspace topology on X F.
We can use these open intervals to define a topology on any ordered set, the order topology. When more than one order is being used on a set one talks about the order topology induced by a particular order. For instance if N is the natural numbers, < is less than and > greater than we might refer to the order topology on N induced by < and the ...
A topology on a set may be defined as the collection of subsets which are considered to be "open". (An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following ...