Free Cells in Hyperspaces of Graphs

• Main Authors: José Ángel Juárez Morales, Gerardo Reyna Hernández, Jesús Romero Valencia, Omar Rosario Cayetano
• Format: Article
• Language: English
• Published: MDPI AG 2021-07-01
• Series: Mathematics
• Subjects: hyperspace; graph; dendroid; dendrite
• Online Access: https://www.mdpi.com/2227-7390/9/14/1627

Often for understanding a structure, other closely related structures with the former are associated. An example of this is the study of hyperspaces. In this paper, we give necessary and sufficient conditions for the existence of finitely-dimensional maximal free cells in the hyperspace <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> of a dendrite <i>G</i>; then, we give necessary and sufficient conditions so that the aforementioned result can be applied when <i>G</i> is a dendroid. Furthermore, we prove that the arc is the unique arcwise connected, compact, and metric space <i>X</i> for which the anchored hyperspace <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is an arc for some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>X</mi></mrow></semantics></math></inline-formula>.