▁topology ▁ar ising ▁from ▁the ▁metric ▁and ▁the ▁one ▁ar ising ▁from ▁the ▁order ▁are ▁identical , ▁but ▁yield ▁different ▁present ations ▁for ▁the ▁topology — in ▁the ▁order ▁topology ▁as ▁ordered ▁intervals , ▁in ▁the ▁metric ▁topology ▁as ▁e psilon - ball s . ▁The ▁D ed ek ind ▁cut s ▁construction ▁uses ▁the ▁order ▁topology ▁presentation , ▁while ▁the ▁C auch y ▁sequences ▁construction ▁uses ▁the ▁metric ▁topology ▁presentation . ▁The ▁re als ▁are ▁a ▁contract ible ▁( h ence ▁connected ▁and ▁simply ▁connected ), ▁separ able ▁and ▁complete ▁metric ▁space ▁of ▁Haus dor ff ▁dimension   1 . ▁The ▁real ▁numbers ▁are ▁locally ▁compact ▁but ▁not ▁compact . ▁There ▁are ▁various ▁properties ▁that ▁uniqu ely ▁specify ▁them ; ▁for ▁instance , ▁all ▁un bounded , ▁connected , ▁and ▁separ able ▁order ▁top ologies ▁are ▁necessarily ▁home omorphic ▁to ▁the ▁re als . ▁ ▁Every ▁non negative ▁real ▁number ▁has ▁a ▁square ▁root ▁in ▁R , ▁although ▁no ▁negative ▁number ▁does . ▁This ▁shows ▁that ▁the ▁order ▁on ▁R ▁is ▁determined ▁by ▁its ▁algebraic ▁structure . ▁Also , ▁every ▁polynomial ▁of ▁odd ▁degree ▁adm its ▁at ▁least ▁one ▁real ▁root : ▁these ▁two ▁properties ▁make ▁R ▁the ▁premier ▁example ▁of ▁a ▁real ▁closed ▁field . ▁Pro ving ▁this ▁is ▁the ▁first ▁half ▁of ▁one ▁proof ▁of ▁the ▁fundamental ▁theorem ▁of ▁algebra . ▁ ▁The ▁re als ▁carry ▁a ▁canonical ▁measure , ▁the ▁Leb es gue ▁measure , ▁which ▁is ▁the ▁H aar ▁measure ▁on ▁their ▁structure ▁as ▁a ▁topological ▁group ▁normal