A convenient category of topological spaces.

Steenrod, N. E.

doi:10.1307/mmj/1028999711

Cited by 381 publications

(236 citation statements)

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“…It was shown in [15] that the topology (T s,X 1 , * ⊗ T s,X 2 , * )/R is homeomorphic to κ(T s,X 2 , * ) is the coarsest topology that is both compactly generated and finer than T…”

Section: Continuity In the Strong Topologymentioning

confidence: 99%

Continuity of channel parameters and operations under various DMC topologies

Nasser

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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We study the continuity of many channel parameters and operations under various topologies on the space of equivalent discrete memoryless channels (DMC). We show that mutual information, channel capacity, Bhattacharyya parameter, probability of error of a fixed code, and optimal probability of error for a given code rate and blocklength, are continuous under various DMC topologies. We also show that channel operations such as sums, products, interpolations, and Arıkan-style transformations are continuous.

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“…It was shown in [15] that the topology (T s,X 1 , * ⊗ T s,X 2 , * )/R is homeomorphic to κ(T s,X 2 , * ) is the coarsest topology that is both compactly generated and finer than T…”

Section: Continuity In the Strong Topologymentioning

confidence: 99%

Continuity of channel parameters and operations under various DMC topologies

Nasser

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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“…Similarly, it has been known since the 1960s that points and open sets are the wrong co-ordinate system for topology. Sheaves in algebraic geometry were based on open sets and not points, whilst algebraic topologists sought more "convenient" categories [Bro64,Ste67], i.e. those that admit general spaces of functions.…”

Section: 4mentioning

confidence: 99%

Foundations for Computable Topology

Taylor¹

2011

The Western Ontario Series in Philosophy of Science

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Foundations should be designed for the needs of mathematics and not vice versa. We propose a technique for doing this that exploits the correspondence between category theory and logic and is potentially applicable to several mathematical disciplines.This method is applied to devising a new paradigm for general topology, called Abstract Stone Duality. We express the duality between algebra and geometry as an abstract monadic adjunction that we turn into a new type theory. To this we add an equation that is satisfied by the Sierpiński space, which plays a key role as the classifier for both open and closed subspaces.In the resulting theory there is a duality between open and closed concepts. This captures many basic properties of compact and closed subspaces, despite the absence of any explicitly infinitary axiom. It offers dual results that link general topology to recursion theory.The extensions and applications of ASD elsewhere that this paper survey include a purely recursive theory of elementary real analysis in which, unlike in previous approaches, the real closed interval [0, 1] in ASD is compact."In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." (Hermann Weyl.)"Point-set topology is a disease from which the human race will soon recover." (Henri Poincaré, quoted in Comic Sections by Des MacHale.)

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“…We will always work in the category of compactly generated spaces (see [30] and [33, I.4]). A spectrum E = {(E(n), σ(n)) | n ∈ Z} is a sequence of pointed spaces {E(n) | n ∈ Z} together with pointed maps σ(n) : E(n) ∧ S 1 → E(n + 1), called structure maps.…”

Section: Review Of Spaces Over a Category And Assembly Mapsmentioning

confidence: 99%

The p-chain Spectral Sequence

Davis¹,

Lueck²

2003

K-Theory

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We introduce a new spectral sequence called the p-chain spectral sequence which converges to the (co-)homology of a contravariant C-space with coefficients in a covariant C-spectrum for a small category C. It is different from the corresponding Atiyah-Hirzebruch type spectral sequence. It can be used in combination with the Isomorphism Conjectures of Baum-Connes and Farrell-Jones to compute algebraic K-and L-groups of group rings and topological K-groups of reduced group C * -algebras.

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A convenient category of topological spaces.

Cited by 381 publications

References 0 publications

Continuity of channel parameters and operations under various DMC topologies

Continuity of channel parameters and operations under various DMC topologies

Foundations for Computable Topology

The p-chain Spectral Sequence

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