itional ▁structure ▁ ▁A ▁finite ▁topological ▁space ▁is ▁pseud omet riz able ▁if ▁and ▁only ▁if ▁it ▁is ▁R 0 . ▁In ▁this ▁case , ▁one ▁possible ▁pseud ometric ▁is ▁given ▁by ▁ ▁where ▁x ▁ ≡ ▁y ▁means ▁x ▁and ▁y ▁are ▁top olog ically ▁ind ist ingu ish able . ▁A ▁finite ▁topological ▁space ▁is ▁met riz able ▁if ▁and ▁only ▁if ▁it ▁is ▁discrete . ▁ ▁Like wise , ▁a ▁topological ▁space ▁is ▁uniform izable ▁if ▁and ▁only ▁if ▁it ▁is ▁R 0 . ▁The ▁uniform ▁structure ▁will ▁be ▁the ▁pseud ometric ▁uniform ity ▁induced ▁by ▁the ▁above ▁pseud ometric . ▁ ▁Al gebra ic ▁topology ▁ ▁Perhaps ▁surprising ly , ▁there ▁are ▁finite ▁topological ▁spaces ▁with ▁n ont rivial ▁fundamental ▁groups . ▁A ▁simple ▁example ▁is ▁the ▁pseud oc irc le , ▁which ▁is ▁space ▁X ▁with ▁four ▁points , ▁two ▁of ▁which ▁are ▁open ▁and ▁two ▁of ▁which ▁are ▁closed . ▁There ▁is ▁a ▁continuous ▁map ▁from ▁the ▁unit ▁circle ▁S 1 ▁to ▁X ▁which ▁is ▁a ▁weak ▁hom ot opy ▁equivalence ▁( i . e . ▁it ▁indu ces ▁an ▁isomorphism ▁of ▁hom ot opy ▁groups ). ▁It ▁follows ▁that ▁the ▁fundamental ▁group ▁of ▁the ▁pseud oc irc le ▁is ▁infinite ▁cy clic . ▁ ▁More ▁generally ▁it ▁has ▁been ▁shown ▁that ▁for ▁any ▁finite ▁abstract ▁simp lic ial ▁complex ▁K , ▁there ▁is ▁a ▁finite ▁topological ▁space ▁X K ▁and ▁a ▁weak ▁hom ot opy ▁equivalence ▁f ▁: ▁| K | ▁→ ▁X K ▁where ▁|