The Experts below are selected from a list of 312 Experts worldwide ranked by ideXlab platform
Zequn Liu - One of the best experts on this subject based on the ideXlab platform.
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Topologies on Quotient space of matrices via semi‐tensor product
Asian Journal of Control, 2019Co-Authors: Daizhan Cheng, Zequn LiuAbstract:An equivalence of matrices via semi-tensor product (STP) is proposed. Using this equivalence, the Quotient space is obtained. Parallel and sequential arrangements of the natural projection on different shapes of matrices lead to the product Topology and Quotient Topology respectively. Then the Frobenious inner product of matrices is extended to equivalence classes, which produces a metric on the Quotient space. This metric leads to a metric Topology. A comparison for these three topologies is presented. Some topological properties are revealed.
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Topologies on Quotient Space of Matrices via Semi-tensor Product
arXiv: Optimization and Control, 2018Co-Authors: Daizhan Cheng, Zequn LiuAbstract:An equivalence of matrices via semi-tensor product (STP) is proposed. Using this equivalence, the Quotient space is obtained. Parallel and sequential arrangements of the natural projection on different shapes of matrices leads to the product Topology and Quotient Topology respectively. Then the Frobenious inner product of matrices is extended to equivalence classes, which produces a metric on the Quotient space. This metric leads to a metric Topology. A comparison for these three topologies is presented. Some topological properties are revealed.
Paul Fabel - One of the best experts on this subject based on the ideXlab platform.
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On zero dimensional sequential spaces
arXiv: General Topology, 2020Co-Authors: Paul FabelAbstract:We develop tools to recognize sequential spaces with large inductive dimension zero. We show the Hawaiian earring group $G$ is 0 dimensional, when endowed with the Quotient Topology, inherited from the space of based loops with the compact open Topology. In particular $G$ is $T_4$ and hence inclusion $G \rightarrow F_M (G)$ is a topological embedding into the free topological group $F_M (G)$ in the sense of Markov.
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On fundamental groups with the Quotient Topology
Journal of Homotopy and Related Structures, 2015Co-Authors: Jeremy Brazas, Paul FabelAbstract:The quasitopological fundamental group $$\pi _{1}^{qtop}(X,x_0)$$ π 1 q t o p ( X , x 0 ) is the fundamental group endowed with the natural Quotient Topology inherited from the space of based loops and is typically non-discrete when $$X$$ X does not admit a traditional universal cover. This topologized fundamental group is an invariant of homotopy type which has the ability to distinguish weakly homotopy equivalent and shape equivalent spaces. In this paper, we clarify various relationships among topological properties of the group $$\pi _{1}^{qtop}(X,x_0)$$ π 1 q t o p ( X , x 0 ) and properties of the underlying space $$X$$ X such as ‘ $$\pi _{1}$$ π 1 -shape injectivity’ and ‘homotopically path-Hausdorff.’ A space $$X$$ X is $$\pi _1$$ π 1 -shape injective if the fundamental group canonically embeds in the first shape group so that the elements of $$\pi _1(X,x_0)$$ π 1 ( X , x 0 ) can be represented as sequences in an inverse limit. We show a locally path connected metric space $$X$$ X is $$\pi _1$$ π 1 -shape injective if and only if $$\pi _{1}^{qtop}(X,x_0)$$ π 1 q t o p ( X , x 0 ) is invariantly separated in the sense that the intersection of all open invariant (i.e. normal) subgroups is the trivial subgroup. In the case that $$X$$ X is not $$\pi _1$$ π 1 -shape injective, the homotopically path-Hausdorff property is useful for distinguishing homotopy classes of loops and guarantees the existence of certain generalized covering maps. We show that a locally path connected space $$X$$ X is homotopically path-Hausdorff if and only if $$\pi _{1}^{qtop}(X,x_0)$$ π 1 q t o p ( X , x 0 ) satisfies the $$T_1$$ T 1 separation axiom.
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On the Topology of quasitopological fundamental groups
arXiv: Algebraic Topology, 2013Co-Authors: Jeremy Brazas, Paul FabelAbstract:We establish and clarify various relationships among topological properties of the fundamental group $\pi_{1}^{qtop}(X,x_0)$ (endowed with a natural Quotient Topology) and properties of the underlying space $X$ such as `homotopically path-Hausdorff' and `$\pi_{1}$-shape injectivity.'
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Compactly generated quasitopological homotopy groups with discontinuous multiplication
arXiv: Algebraic Topology, 2011Co-Authors: Paul FabelAbstract:For each positive integer Q there exists a path connected metric compactum X such that the Qth-homotopy group of X is compactly generated but not a topological group (with the Quotient Topology).
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multiplication is discontinuous in the hawaiian earring group with the Quotient Topology
Bulletin of The Polish Academy of Sciences Mathematics, 2011Co-Authors: Paul FabelAbstract:The natural Quotient map q from the space of based loops in the Hawaiian earring onto the fundamental group provides a new example of a Quotient map such that q x q fails to be a Quotient map. This also settles in the negative the question of whether the fundamental group (with the Quotient Topology) of a compact metric space is always a topological group with the standard operations.
Jeremy Brazas - One of the best experts on this subject based on the ideXlab platform.
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On maps with continuous path lifting
arXiv: Algebraic Topology, 2020Co-Authors: Jeremy Brazas, Atish MitraAbstract:We study a natural generalization of covering projections defined in terms of unique lifting properties. A map $p:E\to X$ has the "continuous path-covering property" if all paths in $X$ lift uniquely and continuously (rel. basepoint) with respect to the compact-open Topology. We show that maps with this property are closely related to fibrations with totally path-disconnected fibers and to the natural Quotient Topology on the homotopy groups. In particular, the class of maps with the continuous path-covering property lies properly between Hurewicz fibrations and Serre fibrations with totally path-disconnected fibers. We extend the usual classification of covering projections to a classification of maps with the continuous path-covering property in terms of topological $\pi_1$: for any path-connected Hausdorff space $X$, maps $E\to X$ with the continuous path-covering property are classified up to weak equivalence by subgroups $H\leq \pi_1(X,x_0)$ with totally path-disconnected coset space $\pi_1(X,x_0)/H$. Here, "weak equivalence" refers to an equivalence relation generated by formally inverting bijective weak homotopy equivalences.
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On the discontinuity of the π1-action
Topology and its Applications, 2018Co-Authors: Jeremy BrazasAbstract:Abstract We show the classical π 1 -action on the n-th homotopy group can fail to be continuous for any n when the homotopy groups are equipped with the natural Quotient Topology. In particular, we prove the action π 1 ( X ) × π n ( X ) → π n ( X ) fails to be continuous for a one-point union X = A ∨ H n where A is an aspherical space such that π 1 ( A ) is a topological group and H n is the ( n − 1 ) -connected, n-dimensional Hawaiian earring space H n for which π n ( H n ) is a topological abelian group.
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GENERALIZED COVERING SPACE THEORIES
arXiv: Algebraic Topology, 2015Co-Authors: Jeremy BrazasAbstract:In this paper, we unify various approaches to generalized covering space theory by introducing a categorical framework in which coverings are dened purely in terms of unique lifting properties. For each categoryC of path-connected spaces having the unit disk as an object, we construct a category of C-coverings over a given space X that embeds in the category of 1(X;x0)-sets via the usual monodromy action on bers. When C is extended to its coreective hull H (C), the resulting category of based H (C)-coverings is complete, has an initial object, and often characterizes more of the subgroup lattice of 1(X;x0) than traditional covering spaces. We apply our results to three special coreective subcategories: (1) The category of -coverings employs the convenient category of -generated spaces and is universal in the sense that it contains every other generalized covering category as a subcategory. (2) In the locally path-connected category, we preserve notion of generalized covering due to Fischer and Zastrow and characterize the Topology of such coverings using the standard whisker Topology. (3) By employing the coreective hull Fan of the category of all contractible spaces, we characterize the notion of continuous lifting of paths and identify the Topology of Fan-coverings as the natural Quotient Topology inherited from the path space.
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On fundamental groups with the Quotient Topology
Journal of Homotopy and Related Structures, 2015Co-Authors: Jeremy Brazas, Paul FabelAbstract:The quasitopological fundamental group $$\pi _{1}^{qtop}(X,x_0)$$ π 1 q t o p ( X , x 0 ) is the fundamental group endowed with the natural Quotient Topology inherited from the space of based loops and is typically non-discrete when $$X$$ X does not admit a traditional universal cover. This topologized fundamental group is an invariant of homotopy type which has the ability to distinguish weakly homotopy equivalent and shape equivalent spaces. In this paper, we clarify various relationships among topological properties of the group $$\pi _{1}^{qtop}(X,x_0)$$ π 1 q t o p ( X , x 0 ) and properties of the underlying space $$X$$ X such as ‘ $$\pi _{1}$$ π 1 -shape injectivity’ and ‘homotopically path-Hausdorff.’ A space $$X$$ X is $$\pi _1$$ π 1 -shape injective if the fundamental group canonically embeds in the first shape group so that the elements of $$\pi _1(X,x_0)$$ π 1 ( X , x 0 ) can be represented as sequences in an inverse limit. We show a locally path connected metric space $$X$$ X is $$\pi _1$$ π 1 -shape injective if and only if $$\pi _{1}^{qtop}(X,x_0)$$ π 1 q t o p ( X , x 0 ) is invariantly separated in the sense that the intersection of all open invariant (i.e. normal) subgroups is the trivial subgroup. In the case that $$X$$ X is not $$\pi _1$$ π 1 -shape injective, the homotopically path-Hausdorff property is useful for distinguishing homotopy classes of loops and guarantees the existence of certain generalized covering maps. We show that a locally path connected space $$X$$ X is homotopically path-Hausdorff if and only if $$\pi _{1}^{qtop}(X,x_0)$$ π 1 q t o p ( X , x 0 ) satisfies the $$T_1$$ T 1 separation axiom.
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On the Topology of quasitopological fundamental groups
arXiv: Algebraic Topology, 2013Co-Authors: Jeremy Brazas, Paul FabelAbstract:We establish and clarify various relationships among topological properties of the fundamental group $\pi_{1}^{qtop}(X,x_0)$ (endowed with a natural Quotient Topology) and properties of the underlying space $X$ such as `homotopically path-Hausdorff' and `$\pi_{1}$-shape injectivity.'
Daizhan Cheng - One of the best experts on this subject based on the ideXlab platform.
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Topologies on Quotient space of matrices via semi‐tensor product
Asian Journal of Control, 2019Co-Authors: Daizhan Cheng, Zequn LiuAbstract:An equivalence of matrices via semi-tensor product (STP) is proposed. Using this equivalence, the Quotient space is obtained. Parallel and sequential arrangements of the natural projection on different shapes of matrices lead to the product Topology and Quotient Topology respectively. Then the Frobenious inner product of matrices is extended to equivalence classes, which produces a metric on the Quotient space. This metric leads to a metric Topology. A comparison for these three topologies is presented. Some topological properties are revealed.
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Topologies on Quotient Space of Matrices via Semi-tensor Product
arXiv: Optimization and Control, 2018Co-Authors: Daizhan Cheng, Zequn LiuAbstract:An equivalence of matrices via semi-tensor product (STP) is proposed. Using this equivalence, the Quotient space is obtained. Parallel and sequential arrangements of the natural projection on different shapes of matrices leads to the product Topology and Quotient Topology respectively. Then the Frobenious inner product of matrices is extended to equivalence classes, which produces a metric on the Quotient space. This metric leads to a metric Topology. A comparison for these three topologies is presented. Some topological properties are revealed.
Rajai Nasser - One of the best experts on this subject based on the ideXlab platform.
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Topological Structures on DMC Spaces
Entropy, 2018Co-Authors: Rajai NasserAbstract:Two channels are said to be equivalent if they are degraded from each other. The space of equivalent channels with input alphabet X and output alphabet Y can be naturally endowed with the Quotient of the Euclidean Topology by the equivalence relation. A Topology on the space of equivalent channels with fixed input alphabet X and arbitrary but finite output alphabet is said to be natural if and only if it induces the Quotient Topology on the subspaces of equivalent channels sharing the same output alphabet. We show that every natural Topology is σ -compact, separable and path-connected. The finest natural Topology, which we call the strong Topology, is shown to be compactly generated, sequential and T 4 . On the other hand, the strong Topology is not first-countable anywhere, hence it is not metrizable. We introduce a metric distance on the space of equivalent channels which compares the noise levels between channels. The induced metric Topology, which we call the noisiness Topology, is shown to be natural. We also study topologies that are inherited from the space of meta-probability measures by identifying channels with their Blackwell measures.
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ISIT - Topological structures on DMC spaces
2017 IEEE International Symposium on Information Theory (ISIT), 2017Co-Authors: Rajai NasserAbstract:Two channels are said to be equivalent if they are degraded from each other. The space of equivalent channels with input alphabet X and output alphabet Y can be naturally endowed with the Quotient of the Euclidean Topology by the equivalence relation. We show that this Topology is compact, path-connected and metrizable. A Topology on the space of equivalent channels with fixed input alphabet X and arbitrary but finite output alphabet is said to be natural if and only if it induces the Quotient Topology on the subspaces of equivalent channels sharing the same output alphabet. We show that every natural Topology is σ-compact, separable and path-connected. On the other hand, if |X| ≥ 2, a Hausdorff natural Topology is not Baire and it is not locally compact anywhere. This implies that no natural Topology can be completely metrized if |X| ≥ 2. The finest natural Topology, which we call the strong Topology, is shown to be compactly generated, sequential and T 4 . On the other hand, the strong Topology is not first-countable anywhere, hence it is not metrizable. We show that in the strong Topology, a subspace is compact if and only if it is rank-bounded and strongly-closed. We provide a necessary and sufficient condition for a sequence of channels to converge in the strong Topology. We introduce a metric distance on the space of equivalent channels which compares the noise levels between channels. The induced metric Topology, which we call the noisiness Topology, is shown to be natural. We also study topologies that are inherited from the space of meta-probability measures by identifying channels with their Blackwell measures. We show that the weak-∗ Topology is exactly the same as the noisiness Topology and hence it is natural. We prove that if |X| ≥ 2, the total variation Topology is not natural nor Baire, hence it is not completely metrizable. Moreover, it is not locally compact anywhere. Finally, we show that the Borel σ-algebra is the same for all Hausdorff natural topologies.
