▁topological ▁invariant , ▁i . e . ▁is ▁preserved ▁by ▁any ▁home omorphism . ▁The ▁F PP ▁is ▁also ▁preserved ▁by ▁any ▁retra ction . ▁ ▁According ▁to ▁Br ou wer ▁fixed ▁point ▁theorem ▁every ▁compact ▁and ▁convex ▁subset ▁of ▁a ▁E uclidean ▁space ▁has ▁the ▁F PP . ▁Comp act ness ▁alone ▁does ▁not ▁imply ▁the ▁F PP ▁and ▁convex ity ▁is ▁not ▁even ▁a ▁topological ▁property ▁so ▁it ▁makes ▁sense ▁to ▁ask ▁how ▁to ▁top olog ically ▁character ize ▁the ▁F PP . ▁In ▁ 1 9 3 2 ▁B ors uk ▁asked ▁whether ▁compact ness ▁together ▁with ▁contract ibility ▁could ▁be ▁a ▁sufficient ▁condition ▁for ▁the ▁F PP ▁to ▁hold . ▁The ▁problem ▁was ▁open ▁for ▁ 2 0 ▁years ▁until ▁the ▁conject ure ▁was ▁dis pro ved ▁by ▁K inos h ita ▁who ▁found ▁an ▁example ▁of ▁a ▁compact ▁contract ible ▁space ▁without ▁the ▁F PP . ▁ ▁References ▁ ▁Category : Fixed ▁points ▁( math emat ics ) <0x0A> </s> ▁Johann a ▁W anka ▁( née ▁Johann a ▁Müller ; ▁born ▁ 1 ▁April ▁ 1 9 5 1 ) ▁is ▁a ▁German ▁politician ▁of ▁the ▁Christian ▁Democratic ▁Union ▁( CD U ) ▁who ▁served ▁as ▁Federal ▁Minister ▁for ▁Education ▁and ▁Research ▁in ▁the ▁government ▁of ▁Chan cell or ▁Ang ela ▁Mer kel ▁from ▁ 2 0 1 3 ▁until ▁ 2 0 1 8 . ▁From ▁ 2 0 0 0 ▁to ▁ 2 0 0 9 , ▁she ▁served ▁as ▁Minister ▁for ▁Science , ▁Research ▁and