scholarly journals Multiplication is Discontinuous in the Hawaiian Earring Group (with the Quotient Topology)

2011 ◽  
Vol 59 (1) ◽  
pp. 77-83 ◽  
Author(s):  
Paul Fabel
Keyword(s):  
Quotient Topology ◽  
Hawaiian Earring

Embeddings of locally compact abelian p-groups in Hawaiian groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yanga Bavuma ◽  
Francesco G. Russo
Keyword(s):  
Algebraic Topology ◽  
Geometric Interpretation ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Hawaiian Earring ◽  
Path Connected

Abstract We show that locally compact abelian p-groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. This result gives a new geometric interpretation for the classification of locally compact abelian groups which are rich in commuting closed subgroups. It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring. Among other things, we find applications for locally compact groups which are just noncompact.

scholarly journals Varieties of topological algebras

1977 ◽  
Vol 23 (2) ◽  
pp. 207-241 ◽  
Author(s):  
Walter Taylor
Keyword(s):  
Proper Class ◽  
Quotient Topology ◽  
Topological Groups ◽  
Topological Algebras ◽  
Fixed Set ◽  
Fixed Type

By a variety of topological algebras we mean a class V of topological algebras of a fixed type closed under the formation of subalgebras, products and quotients (i.e. images under continuous homomorphisms yielding the quotient topology). In symbols, V = SV = PV = QV. if V is also closed under the formation of arbitrary continuous homomorphic images, then V is a wide variety. variety. As an example we have the full variety V = Modr (Σ), the class of all topological algebras of a fixed type τ obeying a fixed set Σ of algebraic identities. But not every wide variety is full, e.g. the class of all indiscrete topological algebras of a fixed type; in fact, as Morris observed (1970b), there exists a proper class of varieties of topological groups.

The Singular Homology of the Hawaiian Earring

2000 ◽  
Vol 62 (1) ◽  
pp. 305-310 ◽  
Author(s):  
Katsuya Eda ◽  
Kazuhiro Kawamura
Keyword(s):  
Singular Homology ◽  
Hawaiian Earring

scholarly journals Topology on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_\infty$

2017 ◽  
Vol 9 (1) ◽  
pp. 22-27 ◽  
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Symmetric Functions ◽  
Complex Banach Space ◽  
Continuous Homomorphism ◽  
The Other ◽  
Quotient Topology ◽  
Abelian Topological Group ◽  
Point Evaluation ◽  
Other Hand ◽  
Continuous Complex ◽  
Complex Valued

It is known that the so-called elementary symmetric polynomials $R_n(x) = \int_{[0,1]}(x(t))^n\,dt$ form an algebraic basis in the algebra of all symmetric continuous polynomials on the complex Banach space $L_\infty,$ which is dense in the Fr\'{e}chet algebra $H_{bs}(L_\infty)$ of all entire symmetric functions of bounded  type on $L_\infty.$ Consequently, every continuous homomorphism $\varphi: H_{bs}(L_\infty) \to \mathbb{C}$ is uniquely determined by the sequence $\{\varphi(R_n)\}_{n=1}^\infty.$ By the continuity of the homomorphism $\varphi,$ the sequence $\{\sqrt[n]{|\varphi(R_n)|}\}_{n=1}^\infty$ is bounded. On the other hand, for every sequence $\{\xi_n\}_{n=1}^\infty \subset \mathbb{C},$ such that the sequence $\{\sqrt[n]{|\xi_n|}\}_{n=1}^\infty$ is bounded,  there exists  $x_\xi \in L_\infty$ such that $R_n(x_\xi) = \xi_n$ for every $n \in \mathbb{N}.$ Therefore, for the point-evaluation functional $\delta_{x_\xi}$ we have $\delta_{x_\xi}(R_n) = \xi_n$ for every $n \in \mathbb{N}.$ Thus, every continuous complex-valued homomorphism of $H_{bs}(L_\infty)$ is a point-evaluation functional at some point of $L_\infty.$ Note that such a point is not unique. We can consider an equivalence relation on $L_\infty,$ defined by $x\sim y \Leftrightarrow \delta_x = \delta_y.$ The spectrum (the set of all continuous complex-valued homomorphisms) $M_{bs}$ of the algebra $H_{bs}(L_\infty)$ is one-to-one with the quotient set $L_\infty/_\sim.$ Consequently, $M_{bs}$ can be endowed with the quotient topology. On the other hand, it is naturally to identify $M_{bs}$ with the set of all sequences $\{\xi_n\}_{n=1}^\infty \subset \mathbb{C}$ such that the sequence $\{\sqrt[n]{|\xi_n|}\}_{n=1}^\infty$ is bounded.We show that the quotient topology is Hausdorffand that $M_{bs}$ with the operation of coordinate-wise addition of sequences forms an abelian topological group.

On Hawaiian Groups of the Infinite Dimensional Hawaiian Earring

2021 ◽  
pp. 107920
Author(s):  
Ameneh Babaee ◽  
Behrooz Mashayekhy ◽  
Hanieh Mirebrahimi ◽  
Hamid Torabi
Keyword(s):  
Infinite Dimensional ◽  
Hawaiian Earring
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