Coarsest topology making certain functions continuous
In general topology and related areas of mathematics, the initial topology (or induced topology[1][2] or weak topology or limit topology or projective topology) on a set with respect to a family of functions on is the coarsest topology on that makes those functions continuous.
The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these.
The dual notion is the final topology, which for a given family of functions mapping to a set is the finest topology on that makes those functions continuous.
^Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
^Adamson, Iain T. (1996). "Induced and Coinduced Topologies". A General Topology Workbook. Birkhäuser, Boston, MA. pp. 23–30. doi:10.1007/978-0-8176-8126-5_3. ISBN 978-0-8176-3844-3. Retrieved July 21, 2020. ... the topology induced on E by the family of mappings ...
general topology and related areas of mathematics, the initialtopology (or induced topology or weak topology or limit topology or projective topology) on...
In mathematics, weak topology is an alternative term for certain initialtopologies, often on topological vector spaces or spaces of linear operators,...
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...
is the initialtopology, which for a given family of functions from a set X {\displaystyle X} into topological spaces is the coarsest topology on X {\displaystyle...
natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which...
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...
an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initialtopology on S. Thus...
coarsest topology may refer to: Initialtopology, the most coarse topology in a certain category of topologies Trivial topology, the most coarse topology possible...
topology. The simplest cases (which nevertheless cover many examples) are the initialtopology and the final topology (Willard (1970)). The initial topology...
a unique initial lift ( A → A i ) I {\displaystyle (A\to A_{i})_{I}} . In Top the initial lift is obtained by placing the initialtopology on the source...
well-behaved with respect to initialtopologies. Specifically, complete regularity is preserved by taking arbitrary initialtopologies and the Tychonoff property...
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X {\displaystyle X} with topology τ {\displaystyle \tau } is a subcollection...
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics....
constructed by placing the initialtopology on the underlying set-theoretic inverse limit. This is known as the limit topology. The set of infinite strings...
Lately, topology control algorithms have been divided into two subproblems: topology construction, in charge of the initial reduction, and topology maintenance...
In topology, two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhoods. That is, if x and y are...
normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively...
areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric...
among topologies, applying speciation (the evolution of species) to preserve innovations, and developing topologies incrementally from simple initial structures...
certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular...
rational numbers. The real numbers form an initial unbounded totally ordered set that is connected in the order topology (defined below). Ordered fields are...
use in routing of mesh networks, specifically pertaining to their initialtopology. It was developed in 1991 by Aaron Kershenbaum, Parviz Kermani, and...
0. The uniform topology generated by the above uniformity is the initialtopology induced by the family C(X). In general, this topology will be coarser...
In topology, a coherent topology is a topology that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent...
set in the analytic topology is the union of a countable collection of unit polydiscs. The Zariski topology is the initialtopology for the affine complex-valued...
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