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Neighborhood base at the identity of free paratopological groups

Ali Sayed Elfard (2013)

Topological Algebra and its Applications

In 1985, V. G. Pestov described a neighborhood base at the identity of free topological groups on a Tychonoff space in terms of the elements of the fine uniformity on the Tychonoff space. In this paper, we extend Postev’s description to the free paratopological groups where we introduce a neighborhood base at the identity of free paratopological groups on any topological space in terms of the elements of the fine quasiuniformity on the space.

Network character and tightness of the compact-open topology

Richard N. Ball, Anthony W. Hager (2006)

Commentationes Mathematicae Universitatis Carolinae

For Tychonoff X and α an infinite cardinal, let α def X : = the minimum number of α  cozero-sets of the Čech-Stone compactification which intersect to X (generalizing -defect), and let rt X : = min α max ( α , α def X ) . Give C ( X ) the compact-open topology. It is shown that τ C ( X ) n χ C ( X ) rt X = max ( L ( X ) , L ( X ) def X ) , where: τ is tightness; n χ is the network character; L ( X ) is the Lindel"of number. For example, it follows that, for X Čech-complete, τ C ( X ) = L ( X ) . The (apparently new) cardinal functions n χ C and rt are compared with several others.

Non-abelian group structure on the Urysohn universal space

Michal Doucha (2015)

Fundamenta Mathematicae

We prove that there exists a non-abelian group structure on the Urysohn universal metric space. More precisely, we introduce a variant of the Graev metric that enables us to construct a free group with countably many generators equipped with a two-sided invariant metric that is isometric to the rational Urysohn space. We list several related open problems.

Nonmetrizable topological dynamical characterization of central sets

Hong-Ting Shi, Hong-Wei Yang (1996)

Fundamenta Mathematicae

Without the restriction of metrizability, topological dynamical systems ( X , T s s G ) are defined and uniform recurrence and proximality are studied. Some well known results are generalized and some new results are obtained. In particular, a topological dynamical characterization of central sets in an arbitrary semigroup (G,+) is given and shown to be equivalent to the usual algebraic characterization.

Note on analytic Moufang loops

Eugen Paal (2004)

Commentationes Mathematicae Universitatis Carolinae

It is explicitly shown how the Lie algebras can be associated with the analytic Moufang loops. The resulting Lie algebra commutation relations are well known from the theory of alternative algebras.

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