On quasi-orbital space
Authors: Hattab Hawete 
Published: April 2017

<p align="left">Let <em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">G </span></em></span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">be a subgroup of the group Homeo(</span></span><em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">E</span></em></span></em><span style="font-family: CMR8; font-size: xx-small;">) of homeomorphisms </span>of a Hausdorff topological space <em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">E</span></em></span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">. The class of an orbit </span></span><em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">O </span></em></span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">of </span></span><em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">G  </span></em></span></em><span style="font-family: CMR8; font-size: xx-small;">is the union of </span>all orbits having the same closure as <em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">O</span></em></span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">. We denote by </span></span><em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">E=</span></em></span></em><span style="font-family: CMEX8; font-size: xx-small;"><span style="font-family: CMEX8; font-size: xx-small;">e</span></span><em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">G  </span></em></span></em><span style="font-family: CMR8; font-size: xx-small;">the space of classes </span>of orbits called quasi-orbit space. A space <em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">X  </span></em></span></em><span style="font-family: CMR8; font-size: xx-small;">is called a quasi-orbital space if </span>it is homeomorphic to <em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">E=</span></em></span></em><span style="font-family: CMEX8; font-size: xx-small;"><span style="font-family: CMEX8; font-size: xx-small;">e</span></span><em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">G </span></em></span></em><span style="font-family: CMR8; font-size: xx-small;"><span style="font-family: CMR8; font-size: xx-small;">where </span></span><em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">E  </span></em></span></em><span style="font-family: CMR8; font-size: xx-small;">is a compact Hausdorff space. In this </span>paper, we show that every in nite second countable quasi-compact <em><span style="font-family: CMMI8; font-size: xx-small;"><em><span style="font-family: CMMI8; font-size: xx-small;">T</span></em></span></em><span style="font-family: CMR6; font-size: xx-small;"><span style="font-family: CMR6; font-size: xx-small;">0</span></span><span style="font-family: CMR8; font-size: xx-small;">-space </span>is the quotient of a quasi-orbital space.</p><p align="left"> </p><p align="left"> </p>