scholarly journals Measuring in Weighted Environments: Moving from Metric to Order Topology (Knowing When Close Really Means Close)

10.5772/63670 ◽  
2016 ◽  
Author(s):  
Claudio Garuti
Keyword(s):  
Order Topology

scholarly journals Non-Hermitian route to higher-order topology in an acoustic crystal

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
He Gao ◽  
Haoran Xue ◽  
Zhongming Gu ◽  
Tuo Liu ◽  
Jie Zhu ◽  
...  
Keyword(s):  
Higher Order ◽  
Order Topology ◽  
Topological Phases ◽  
Edge States ◽  
Experimental Realization ◽  
Artificial Materials ◽  
Intrinsic Loss ◽  
Topological Phases Of Matter ◽  
Hierarchical Features ◽  
Nontrivial Topology

AbstractTopological phases of matter are classified based on their Hermitian Hamiltonians, whose real-valued dispersions together with orthogonal eigenstates form nontrivial topology. In the recently discovered higher-order topological insulators (TIs), the bulk topology can even exhibit hierarchical features, leading to topological corner states, as demonstrated in many photonic and acoustic artificial materials. Naturally, the intrinsic loss in these artificial materials has been omitted in the topology definition, due to its non-Hermitian nature; in practice, the presence of loss is generally considered harmful to the topological corner states. Here, we report the experimental realization of a higher-order TI in an acoustic crystal, whose nontrivial topology is induced by deliberately introduced losses. With local acoustic measurements, we identify a topological bulk bandgap that is populated with gapped edge states and in-gap corner states, as the hallmark signatures of hierarchical higher-order topology. Our work establishes the non-Hermitian route to higher-order topology, and paves the way to exploring various exotic non-Hermiticity-induced topological phases.

On a theorem of Fleischer

1986 ◽  
Vol 40 (2) ◽  
pp. 261-266 ◽  
Author(s):  
G. Mehta
Keyword(s):  
Ordered Set ◽  
Separation Theorem ◽  
Order Topology ◽  
Topological Spaces ◽  
Real Numbers ◽  
The Real ◽  
Linearly Ordered Set

AbstractFleischer proved that a linearly ordered set that is separable in its order topology and has countably many jumps is order-isomorphic to a subset of the real numbers. The object of this paper is to extend Fleischer's result and to prove it in a different way. The proof of the theorem is based on Nachbin's extension to ordered topological spaces of Urysohn's separation theorem in normal topological spaces.

scholarly journals Three-dimensional superconductors with hybrid higher-order topology

2019 ◽  
Vol 99 (12) ◽  
Author(s):  
Nick Bultinck ◽  
B. Andrei Bernevig ◽  
Michael P. Zaletel
Keyword(s):  
Three Dimensional ◽  
Higher Order ◽  
Order Topology

scholarly journals Intermediate values and inverse functions on non-Archimedean fields

2002 ◽  
Vol 30 (3) ◽  
pp. 165-176 ◽  
Author(s):  
Khodr Shamseddine ◽  
Martin Berz
Keyword(s):  
Closed Interval ◽  
Inverse Function ◽  
Order Topology ◽  
Primitive Function ◽  
Mild Conditions ◽  
Inverse Functions ◽  
Topological Sense

Continuity or even differentiability of a function on a closed interval of a non-Archimedean field are not sufficient for the function to assume all the intermediate values, a maximum, a minimum, or a unique primitive function on the interval. These problems are due to the total disconnectedness of the field in the order topology. In this paper, we show that differentiability (in the topological sense), together with some additional mild conditions, is indeed sufficient to guarantee that the function assumes all intermediate values and has a differentiable inverse function.

On topological spaces equivalent to ordinals

1988 ◽  
Vol 53 (3) ◽  
pp. 785-795 ◽  
Author(s):  
Jörg Flum ◽  
Juan Carlos Martinez
Keyword(s):  
Model Theory ◽  
Classical Model ◽  
Order Topology ◽  
Topological Spaces ◽  
Locally Compact

AbstractLet L be one of the topological languages Lt, (L∞ω)t and (Lκω)t. We characterize the topological spaces which are models of the L-theory of the class of ordinals equipped with the order topology. The results show that the role played in classical model theory by the property of being well-ordered is taken over in the topological context by the property of being locally compact and scattered.

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