We characterize the topology of the Glimm space of a separable thmlsrc">text stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000562&_mathId=si1.gif&_user=111111111&_pii=S0022123616000562&_rdoc=1&_issn=00221236&md5=bab37beae6480f80d738bc0e4fc9b7c6" title="Click to view the MathML source">C⁎thContainer hidden">thCode">th altimg="si1.gif" overflow="scroll">C⁎th>-algebra and extend the main result of [7] to non-unital AF thmlsrc">text stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616000562&_mathId=si1.gif&_user=111111111&_pii=S0022123616000562&_rdoc=1&_issn=00221236&md5=bab37beae6480f80d738bc0e4fc9b7c6" title="Click to view the MathML source">C⁎thContainer hidden">thCode">th altimg="si1.gif" overflow="scroll">C⁎th>-algebras.