▁components ▁of ▁a ▁finite ▁topological ▁space ▁are ▁precisely ▁the ▁( weak ly ) ▁connected ▁components ▁of ▁the ▁associated ▁graph ▁ Γ . ▁That ▁is , ▁there ▁is ▁a ▁topological ▁path ▁from ▁x ▁to ▁y ▁if ▁and ▁only ▁if ▁there ▁is ▁an ▁und irect ed ▁path ▁between ▁the ▁corresponding ▁vertices ▁of ▁ Γ . ▁ ▁Every ▁finite ▁space ▁is ▁locally ▁path - connected ▁since ▁the ▁set ▁ ▁is ▁a ▁path - connected ▁open ▁neighborhood ▁of ▁x ▁that ▁is ▁contained ▁in ▁every ▁other ▁neighborhood . ▁In ▁other ▁words , ▁this ▁single ▁set ▁forms ▁a ▁local ▁base ▁at ▁x . ▁ ▁Therefore , ▁a ▁finite ▁space ▁is ▁connected ▁if ▁and ▁only ▁if ▁it ▁is ▁path - connected . ▁The ▁connected ▁components ▁are ▁precisely ▁the ▁path ▁components . ▁Each ▁such ▁component ▁is ▁both ▁closed ▁and ▁open ▁in ▁X . ▁ ▁Fin ite ▁spaces ▁may ▁have ▁stronger ▁connect ivity ▁properties . ▁A ▁finite ▁space ▁X ▁is ▁hyper connected ▁if ▁and ▁only ▁if ▁there ▁is ▁a ▁greatest ▁element ▁with ▁respect ▁to ▁the ▁special ization ▁pre order . ▁This ▁is ▁an ▁element ▁whose ▁closure ▁is ▁the ▁whole ▁space ▁X . ▁ult ra connected ▁if ▁and ▁only ▁if ▁there ▁is ▁a ▁least ▁element ▁with ▁respect ▁to ▁the ▁special ization ▁pre order . ▁This ▁is ▁an ▁element ▁whose ▁only ▁neighborhood ▁is ▁the ▁whole ▁space ▁X . ▁For ▁example , ▁the ▁particular ▁point ▁topology ▁on ▁a ▁finite ▁space ▁is ▁hyper connected ▁while ▁the ▁ex cluded ▁point ▁topology ▁is ▁ult ra connected . ▁The ▁S ier pi ński ▁space ▁is ▁both . ▁ ▁Add