In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to:
the final topology on the disjoint union
the topology arising from a norm
the strong operator topology
the strong topology (polar topology), which subsumes all topologies above.
A topology τ is stronger than a topology σ (is a finer topology) if τ contains all the open sets of σ.
In algebraic geometry, it usually means the topology of an algebraic variety as complex manifold or subspace of complex projective space, as opposed to the Zariski topology (which is rarely even a Hausdorff space).
mathematics, a strongtopology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending...
or given topology (the reader is cautioned against using the terms "initial topology" and "strongtopology" to refer to the original topology since these...
analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on...
equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of X , {\displaystyle X,} where this topology is denoted by...
Then the topology τ1 is said to be a coarser (weaker or smaller) topology than τ2, and τ2 is said to be a finer (stronger or larger) topology than τ1....
stronger than all topologies below. The σ-strong-* topology or ultrastrong-* topology is the weakest topologystronger than the ultrastrong topology such...
areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric...
Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the part of mathematics concerned with the properties of a geometric object...
ultrastrong topology, or σ-strongtopology, or strongest topology on the set B(H) of bounded operators on a Hilbert space is the topology defined by the...
with the topology of uniform convergence on bounded sets Strongtopology (polar topology) – Continuous dual space endowed with the topology of uniform...
Network topology is the arrangement of the elements (links, nodes, etc.) of a communication network. Network topology can be used to define or describe...
are the three most important special cases. The strongtopology on V ′ {\displaystyle V'} is the topology of uniform convergence on bounded subsets in V...
related areas of mathematics a polar topology, topology of G {\displaystyle {\mathcal {G}}} -convergence or topology of uniform convergence on the sets...
space topology of uniform convergence on some sub-collection of bounded subsets StrongtopologyTopologies on spaces of linear maps Weak topology – Mathematical...
general topology topics. Topological space Topological property Open set, closed set Clopen set Closure (topology) Boundary (topology) Dense (topology) G-delta...
norm topology. For B(H) = M(K(H)), the strict topology is the σ-strong* topology. It follows from above that B(H) is complete in the σ-strong* topology. Let...
algebraically, differential topology has strong links to algebraic topology. The central goal of the field of differential topology is the classification of...
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H...
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics....
with the topology of uniform convergence on bounded subsets of X {\displaystyle X} ; this topology is also called the strong dual topology and it is...
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...
has two natural topologies: weak and strong (Hirsch 1997). When the manifold is compact, these two topologies agree. The weak topology is always metrizable...
{\displaystyle Y} , the projective topology, or π-topology, on X ⊗ Y {\displaystyle X\otimes Y} is the strongest topology which makes X ⊗ Y {\displaystyle...
\left(C_{c}^{\infty }(U)\right)^{\prime },} is endowed with the strong dual topology. There are other possible choices for the space of test functions...
In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set X , {\displaystyle...
(resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive...
also strongly convergent (Schur 1921). However, since the weak topology on infinite-dimensional spaces is strictly weaker than the strongtopology, there...
compactification, ending up with a compact manifold (for the strongtopology, rather than the Zariski topology, that is). At about the same time as Hironaka's work...
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