Cones That Are 1/2-Homogeneous

• Autores: Sam B., Jr. Nadler, Patricia Pellicer Covarrubias
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 33, Nº 1, 2007, págs. 229-247
• Idioma: inglés
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• Resumen

  • A space is 1/2-homogeneous provided that there are exactly two orbits for the action of the group of homeomorphisms of the space onto itself. Let X be a nonempty compact metric space such that the cone over X is 1/2-homogeneous. It is shown that if X is finite-dimensional, then X is an absolute neighborhood retract. A general theorem is proved which shows that finite dimensionality is necessary. It is shown that if X is a 1-dimensional continuum or if X does not contain certain types of triods in some nonempty open set, then X is an arc or a simple closed curve (assuming Cone(X) is 1/2-homogeneous). A number of corollaries are derived. Some unanswered questions are stated.