Abstract
This paper surveys topologies, called admissible group topologies, of the full group of self-homeomorphisms <svg style="vertical-align:-2.3205pt;width:42.299999px;" id="M1" height="15.0875" version="1.1" viewBox="0 0 42.299999 15.0875" width="42.299999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x210B" d="M702 678l12 -14l-18 -14q-124 -96 -232 -310q25 10 62 21l53 18l33 43q101 126 187 195q88 70 135 70q57 0 57 -50q0 -78 -98 -145q-86 -58 -206 -105q-174 -251 -174 -334q0 -15 9.5 -25t24.5 -10q55 0 167 129l21 -13q-120 -146 -202 -146q-39 0 -63 24t-24 65
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q-110 -50 -165.5 -98t-55.5 -99q0 -24 24 -29q67 0 197 226z" /></g><g transform="matrix(.017,-0,0,-.017,17.01,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,22.892,12.138)"><path id="x1D44B" d="M775 650l-6 -28q-60 -6 -81.5 -16t-61.5 -54l-175 -191l125 -243q30 -58 48.5 -71t82.5 -19l-5 -28h-275l7 28l35 4q31 4 37 12t-6 34l-108 216q-140 -165 -177 -219q-16 -22 -10.5 -30.5t41.5 -13.5l22 -3l-7 -28h-244l8 28q52 4 75 15.5t67 52.5q48 46 206 231
l-110 215q-26 51 -44 63t-72 17l6 28h250l-6 -28l-27 -4q-30 -5 -35 -10t3 -27q17 -43 95 -190q70 78 154 185q15 21 10 29.5t-33 12.5l-30 4l5 28h236z" /></g><g transform="matrix(.017,-0,0,-.017,36.355,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg> of a Tychonoff space <svg style="vertical-align:-0.0pt;width:13.5875px;" id="M2" height="11.175" version="1.1" viewBox="0 0 13.5875 11.175" width="13.5875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44B"/></g> </svg>, which yield continuity of both the group operations and at the same time provide continuity of the evaluation function or, in other words, make the evaluation function a group action of <svg style="vertical-align:-2.3205pt;width:42.299999px;" id="M3" height="15.0875" version="1.1" viewBox="0 0 42.299999 15.0875" width="42.299999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x210B"/></g><g transform="matrix(.017,-0,0,-.017,17.01,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,22.892,12.138)"><use xlink:href="#x1D44B"/></g><g transform="matrix(.017,-0,0,-.017,36.355,12.138)"><use xlink:href="#x29"/></g> </svg> on <svg style="vertical-align:-0.0pt;width:13.5875px;" id="M4" height="11.175" version="1.1" viewBox="0 0 13.5875 11.175" width="13.5875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44B"/></g> </svg>. By means of a compact extension procedure, beyond local compactness and in two essentially different cases of rim-compactness, we show that the complete upper-semilattice <svg style="vertical-align:-3.27605pt;width:52.487499px;" id="M5" height="16.275" version="1.1" viewBox="0 0 52.487499 16.275" width="52.487499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.137)"><path id="x2112" d="M608 643l-12 -19q-56 33 -145 33q-87 0 -154 -42q-68 -42 -68 -115q0 -74 66 -116q67 -42 176 -42q21 0 63 5l29 56q34 66 57 103q43 65 106 113q65 50 122 50q41 0 62 -22t21 -60q0 -67 -87 -144q-88 -76 -217 -110q-46 -101 -119 -186q-72 -83 -136 -108
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h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" /></g> <g transform="matrix(.017,-0,0,-.017,27.2,12.137)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,33.082,12.137)"><use xlink:href="#x1D44B"/></g><g transform="matrix(.017,-0,0,-.017,46.545,12.137)"><use xlink:href="#x29"/></g> </svg> of all admissible group topologies on <svg style="vertical-align:-2.3205pt;width:42.299999px;" id="M6" height="15.0875" version="1.1" viewBox="0 0 42.299999 15.0875" width="42.299999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x210B"/></g><g transform="matrix(.017,-0,0,-.017,17.01,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,22.892,12.138)"><use xlink:href="#x1D44B"/></g><g transform="matrix(.017,-0,0,-.017,36.355,12.138)"><use xlink:href="#x29"/></g> </svg> admits a least element, that can be described simply as a set-open topology and contemporaneously as a uniform topology. But, then, carrying on another efficient way to produce admissible group topologies in substitution of, or in parallel with, the compact extension procedure, we show that rim-compactness is not a necessary condition for the existence of the least admissible group topology. Finally, we give necessary and sufficient conditions for the topology of uniform convergence on the bounded sets of a local proximity space to be an admissible group topology. Also, we cite that local compactness of <svg style="vertical-align:-0.0pt;width:13.5875px;" id="M7" height="11.175" version="1.1" viewBox="0 0 13.5875 11.175" width="13.5875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44B"/></g> </svg> is not a necessary condition for the compact-open topology to be an admissible group topology of <svg style="vertical-align:-2.3205pt;width:42.299999px;" id="M8" height="15.0875" version="1.1" viewBox="0 0 42.299999 15.0875" width="42.299999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x210B"/></g><g transform="matrix(.017,-0,0,-.017,17.01,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,22.892,12.138)"><use xlink:href="#x1D44B"/></g><g transform="matrix(.017,-0,0,-.017,36.355,12.138)"><use xlink:href="#x29"/></g> </svg>.
Highlights
The “incipit” of the homeomorphism group theory resides in the early seminal work of Birkoff [1]
With an apparent simplicity joined with an impressive bright proof strategy, Birkoff positively answered to the query: there exists a topology on the full self-homeomorphism group H(X) of a compact metric space X which makes it into a topological group and a subspace of the Hilbert cube? The area, originating from [1], has initially evolved relaxing the compactness condition by passing from the class of compact metric spaces, as in Birkoff, to the class of T2 locally compact spaces, as in Arens [2]
Birkoff and Arens, we focused our investigation on topologies which make H(X) a topological group and the evaluation function a group action of H(X) on X and, rather obviously, looked at uniform topologies
Summary
This paper surveys topologies, called admissible group topologies, of the full group of self-homeomorphisms H(X) of a Tychonoff space X, which yield continuity of both the group operations and at the same time provide continuity of the evaluation function or, in other words, make the evaluation function a group action of H(X) on X. By means of a compact extension procedure, beyond local compactness and in two essentially different cases of rim-compactness, we show that the complete upper-semilattice LH(X) of all admissible group topologies on H(X) admits a least element, that can be described as a set-open topology and contemporaneously as a uniform topology. Carrying on another efficient way to produce admissible group topologies in substitution of, or in parallel with, the compact extension procedure, we show that rim-compactness is not a necessary condition for the existence of the least admissible group topology. We give necessary and sufficient conditions for the topology of uniform convergence on the bounded sets of a local proximity space to be an admissible group topology. We cite that local compactness of X is not a necessary condition for the compact-open topology to be an admissible group topology of H(X)
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