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An asymmetric Ellis theorem

Journal Article


Abstract


  • In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology T making all translations continuous also has

    jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem does not apply to locally

    compact asymmetric spaces such as the reals with addition and the topology of upper open rays.We first show a bitopological Ellis

    theorem, and then introduce a generalization of locally compact Hausdorff, called locally skew compact, and a topological dual,

    T k, to obtain the following asymmetric Ellis theorem which applies to the example above:

    Whenever (X, ÃÂÃÂÃÂ÷,T ) is a group with a locally skew compact topology making all translations continuous, then multiplication

    is jointly continuous in both (X, ÃÂÃÂÃÂ÷,T ) and (X, ÃÂÃÂÃÂ÷,T k), and inversion is a homeomorphism between (X,T ) and (X,T k).

    This generalizes the classical Ellis theorem, because T = T k when (X,T ) is locally compact Hausdorff.

Authors


  •   Andima, Susan (external author)
  •   Kopperman, Ralph (external author)
  •   Nickolas, Peter R.

Publication Date


  • 2007

Citation


  • Andima, S., Kopperman, R. & Nickolas, P. R. (2007). An asymmetric Ellis theorem. Topology and Its Applications: a journal devoted to general, geometric, set-theoretic and algebraic, 155 146-160.

Scopus Eid


  • 2-s2.0-36248937774

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/3045

Number Of Pages


  • 14

Start Page


  • 146

End Page


  • 160

Volume


  • 155

Place Of Publication


  • www.elsevier.com/locate/topol

Abstract


  • In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology T making all translations continuous also has

    jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem does not apply to locally

    compact asymmetric spaces such as the reals with addition and the topology of upper open rays.We first show a bitopological Ellis

    theorem, and then introduce a generalization of locally compact Hausdorff, called locally skew compact, and a topological dual,

    T k, to obtain the following asymmetric Ellis theorem which applies to the example above:

    Whenever (X, ÃÂÃÂÃÂ÷,T ) is a group with a locally skew compact topology making all translations continuous, then multiplication

    is jointly continuous in both (X, ÃÂÃÂÃÂ÷,T ) and (X, ÃÂÃÂÃÂ÷,T k), and inversion is a homeomorphism between (X,T ) and (X,T k).

    This generalizes the classical Ellis theorem, because T = T k when (X,T ) is locally compact Hausdorff.

Authors


  •   Andima, Susan (external author)
  •   Kopperman, Ralph (external author)
  •   Nickolas, Peter R.

Publication Date


  • 2007

Citation


  • Andima, S., Kopperman, R. & Nickolas, P. R. (2007). An asymmetric Ellis theorem. Topology and Its Applications: a journal devoted to general, geometric, set-theoretic and algebraic, 155 146-160.

Scopus Eid


  • 2-s2.0-36248937774

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/3045

Number Of Pages


  • 14

Start Page


  • 146

End Page


  • 160

Volume


  • 155

Place Of Publication


  • www.elsevier.com/locate/topol