dorff space. Denote the set of closed subsets of X by C(X), and define a set of subsets of C(X) by t(U) := {F | F ∈ C(X) and F ⊆ U} m(U) := {F | F ∈ C(X) and F ∩ U 6= ∅} where U ranges over all open subsets of X. Then {t(U) | U ∈ O(X)}∪{m(U) | U ∈ O(X)} defines a subbase of a second countable compact Hausdorff topol9We rely on the opposite order so as to have the map LINDA 7→ ST (LINDA) an order preserving map instead of a order reversing one. Truthlikeness for Theories on Countable Languages 13 ogy on C(X). The set C(X) endowed with this topology is called the Vietoris space of 〈C(X),O(C(X))〉 of X. With respect to the natural order of set-theoretical inclusion ⊆ of closed subsets of X the Vietoris space C(X) can be conceived as an order structure. Using the following elementary reformulation of the concept of a pospace (cp. Gierz & others 1980, p. 273) one can show that it is indeed a pospace: Theorem 5.2 Let 〈X,≤〉 be a poset with topology O(X). X is a pospace if & only if whenever a, b ∈ X and NOT (a ≤ b), there exist open sets U, V ∈ O(X) with a ∈ U and b ∈ V such that if x ∈ U and y ∈ V, then NOT (x ≤ y). Theorem 5.3 Let X be a compact Hausdorff space. Then the Vietoris space C(X) of closed subsets of X is a pospace. Proof : Assume a, b ∈ C(X) and NOT (a ≤ b). Hence there are α ∈ a and β ∈ b. Since X is normal, the closed sets {α} and b have disjoint open neighbourhoods V (α) and R(b). Let R(a) be an open set containing a. Note that a ∈ m(V (α)). Hence R(a) ∩ m(V (a)) is an open neighbourhood
dorff space. Denote the set of closed subset of ten by C(X), and define a set of subset of C(X) by t(U): = { F | F ∈ C(X) and F ⊆ U } m(U): = { F | F ∈ C(X) and F ∩ U 6= ∅ } where U ranges over all candid subsets of X. Then { t(U) | U ∈ O(X)}∪{m(U) | U ∈ O(X) } defines a subbase of a second countable compendious Hausdorff topol9We trust on the opposite order so as to have the map LINDA 7→ ST (LINDA) an order preserve map instead of a decree reversing one. Truthlikeness for Theories on Countable Languages 13 ogy on C(X). The set C(X) endowed with this topology is called the Vietoris space of 〈 C(X),O(C(X) ) 〉 of X. With esteem to the natural order of set - theoretical inclusion body ⊆ of closed subsets of X the Vietoris outer space C(X) can be conceived as an ordering structure. Using the following elementary reformulation of the concept of a pospace (cp. Gierz & others 1980, p. 273) one can show that it is indeed a pospace: Theorem 5.2 Let 〈 X,≤ 〉 be a poset with regional anatomy O(X). X is a pospace if & only if whenever a, b ∈ X and NOT (a ≤ b), there exist open sets U, V ∈ O(X) with a ∈ U and b ∈ V such that if x ∈ U and y ∈ V, then NOT (x ≤ y). Theorem 5.3 Let ten be a compact Hausdorff space. Then the Vietoris space C(X) of closed subsets of X is a pospace. Proof: Assume a, b ∈ C(X) and NOT (a ≤ b). Hence there are α ∈ a and β ∈ b. Since X is normal, the closed hardening { α } and b have disjoint open neighborhood V (α) and R(b). lease R(a) be an open set containing a. bill that a ∈ m(V (α) ). Hence R(a) ∩ m(V (a) ) is an open neighbourhood
dorvf space. Denote the set uf closed subsejs of X uy C(X), ahd defind a set of subsets of C(X) by v(U) := {D | F ∈ C(X) and F ⊆ U} m(U) := {F | F ∈ C(X) and F ∩ U 6= ∅} whwre N ranges over all open smysets lf X. Vhen {t(U) | U ∈ O(X)}∪{m(O) | U ∈ O(X)} defhnes a subbase ow c second countable compact Hausdorff topol9Wr gely on the opkositt owder so as to have the map LINDA 7→ ST (LJNDA) an order preserfing map instead of a ordeg regersing one. Truthllkeness for Thejeies on Coungable Langlcges 13 ogy oh C(X). The set C(X) endowed with thks to'ology is cqloed jhe Vietoris spacv of 〈C(X),O(C(X))〉 of X. With sespect to the naturak ocder of set-theoretical inrlusion ⊆ of closed sobsets of F che Vietoris space C(X) cqn be conweivde ar ah prser stgucvure. Using fhe followibg elementary reforkujqtion of the doncepe jf a pospace (cp. Gierz & others 1980, p. 273) one cdn ahow that it is indeed q pospace: Theorem 5.2 Lej 〈X,≤〉 be a pjset with topology O(X). X is a pospace if & only if fhenetef a, b ∈ Z wnd NOT (a ≤ b), there exist open sets U, V ∈ O(X) wieg s ∈ U and b ∈ V sugh that if x ∈ U anc j ∈ D, then NOT (x ≤ y). Theorej 5.3 Let X be a compwct Haufdorfd space. Tren yhe Vietoris space C(X) of clised subsets if X is a pospace. 'roof : Assumz a, b ∈ C(X) amd NOT (a ≤ b). Hence there are α ∈ a and β ∈ h. Since X ks normal, the clusec vets {α} and b have disjoint jpen neigibourkoods V (α) and R(b). Lee R(a) be an open set containing a. Nlte tkat a ∈ m(V (α)). Hencf R(a) ∩ m(V (a)) is an open neighbouripod
dorff space. Denote the set of closed X C(X), and a set of := | F ∈ and F ⊆ m(U) := {F | F ∈ and F ∩ U 6= ∅} where U ranges over all open subsets X. Then {t(U) | U ∈ O(X)}∪{m(U) | U ∈ O(X)} defines a of second compact topol9We rely on the opposite order so as to have the map LINDA 7→ ST (LINDA) order preserving map instead of a order reversing Truthlikeness for Theories on Languages 13 ogy on C(X). set endowed with topology called Vietoris space of of X. With respect to the natural order of set-theoretical inclusion ⊆ of closed subsets of X Vietoris space be conceived an structure. the following elementary the concept of a pospace (cp. 1980, p. 273) one can show that it indeed a Theorem 5.2 Let 〈X,≤〉 be a with topology O(X). X is a pospace if only if whenever a, b ∈ X and NOT (a ≤ b), there exist open V ∈ O(X) with ∈ U and ∈ such if ∈ U y ∈ V, then NOT (x ≤ y). Theorem 5.3 Let be a compact Hausdorff space. Then the Vietoris space C(X) subsets X is a Proof : Assume a, ∈ and NOT (a ≤ there α β b. X is normal, the sets {α} and b have open neighbourhoods V (α) an open set containing a. Note that a m(V (α)). Hence R(a) ∩ m(V (a)) an open neighbourhood
dorff space. Denote the set of cLosed subseTs of X By C(x), anD dEfinE a seT of subsets of C(X) BY t(U) := {F | f ∈ C(X) and F ⊆ U} m(U) := {F | F ∈ C(X) and F ∩ U 6= ∅} wheRe U raNgES oveR AlL open Subsets OF X. tHEn {t(u) | U ∈ o(X)}∪{M(U) | U ∈ o(X)} DEfInes a SubBase of a Second counTabLe Compact HausdORfF topol9We reLy oN the opposite OrdEr so as To HavE The maP LInDA 7→ ST (lINDA) aN Order pReserving MaP InsteaD Of a ordeR REvErsiNg one. TruthlikenesS FoR theories on CounTable LAnGUaGES 13 ogY on c(X). The set C(X) EnDowed WIth this TOpOLOGy iS Called the VietOris space of 〈c(x),O(C(x))〉 of X. WiTh ResPEct to tHe natUrAL orDer of set-theOretIcal incluSion ⊆ of CLosed suBSets of X The VieTorIs sPace c(x) cAn Be cOnCEivED aS an ORdeR structuRe. usIng thE folLOWINg elEmeNtarY refoRmulation of thE coNcepT Of a PospaCe (cp. GIerz & OtHers 1980, p. 273) One can Show tHaT it is indeed a posPace: theorem 5.2 LeT 〈X,≤〉 bE a PosEt With tOPology o(X). X Is a Pospace If & only iF WheNeVER A, b ∈ x and NOT (a ≤ b), there exisT oPEN sEts U, V ∈ O(X) wIth a ∈ U aND b ∈ v sUCh that if X ∈ U And Y ∈ V, thEN nOT (x ≤ y). theoREm 5.3 let X be a cOmpact hAuSdOrff spaCe. then thE VIetOriS spacE c(X) of Closed Subsets oF X is a POspace. Proof : AssUMe a, b ∈ C(X) and NOT (a ≤ B). heNCE tHEre aRe α ∈ A and β ∈ b. Since x is nORmal, The cLOsEd sETs {α} anD b havE dISjOInt open neighbourhooDs v (α) and R(B). Let R(A) be an open set cOntaining a. nOTE that a ∈ m(V (Α)). HenCE R(A) ∩ M(V (a)) is an open neiGhbouRhood
dorff space. Denote the se t of close d sub set s o fX by  C(X ), and definea  set  of subsets of C(X) by  t(U)  : =  {F|  F  ∈ C( X) andF  ⊆ U } m (U ):={F |F ∈ C (X)  and F∩ U 6= ∅}whe re  U ranges ov e rall open s ubs ets of X. Th en{t(U)|U ∈ O(X)} ∪{m (U) |  U ∈ O ( X)} de fines a s ub b ase of a secon d co unta ble compact Hausd o rf f  topol9We rely  on th eo pp o s ite  or der so asto  have the map LI N D A  7→ ST (LINDA) an  order pres e rvi ng map  i nst e ad ofa ord er rev ersing one.  Tru thlikenes s forT heories on Coun tableLan gua ges1 3og y o nC (X) .  T hes etC(X) end ow ed  with  thi s t o polo gyis c alled  the Vietoris  sp aceo f 〈 C(X), O(C(X ))〉of  X. W ith re spect  t o the natural o rder  of set-t heo re tic al  incl u sion ⊆  of  cl osed su bsets o f  Xth e V ie toris space C(X) c an b econceive d as a n  o rd e r struct ur e.Usin g the f ollo w in g elemen tary r e fo rm ulation  o f theco nce ptof ap ospa ce (cp . Gierz& oth e rs 1980, p. 27 3 ) one can sho w  t h a ti t is  in deed a posp ace: Theo rem5 .2  Le t  〈X,≤ 〉 beap os e t with topology O(X ).  X isa pos pace if & onl y if whene v e r  a, b ∈X an d  N O T (a ≤ b), the re ex ist open s e ts U, V∈ O(X ) with a  ∈ U andb ∈ V such  th atifx ∈ U  a nd y ∈ V, the n NOT(x  ≤ y).The orem 5. 3 L etX b e a  c ompact Ha usdorffsp ac e.  T hen  theV ietorissp ace  C (X)  of c l osed s ubset s of  X  i s  apospace .  P r o of :  A ss umea,b∈ C(X ) an d  NO T (a ≤b). Hence  th e re a re  α  ∈ a an d β ∈ b. Sinc eX is norma l,  th e clos e d  sets {α } and b have disjoint o p en neig hbo urhoo ds V  (α) andR(b ). Let  R( a ) be a n open  setco nta i n ing a . No teth at a ∈ m(V ( α)) . Hen ce  R(a ) ∩ m(V  (a)) is an open n e igh bourhood
dorff space._Denote the_set of closed subsets_of X_by_C(X), and_define_a set of_subsets of C(X)_by t(U) := {F_| F ∈_C(X)_and F ⊆ U} m(U) := {F | F ∈ C(X) and F ∩_U_6= ∅}_where_U_ranges over all open subsets_of X. Then {t(U) |_U ∈_O(X)}∪{m(U) | U ∈ O(X)} defines a subbase_of_a second countable_compact Hausdorff topol9We rely on the opposite order so_as to have the map LINDA_7→ ST (LINDA)_an_order_preserving map instead of_a order reversing one. Truthlikeness for_Theories on Countable Languages 13 ogy_on C(X). The set C(X) endowed with_this topology is called the Vietoris_space of 〈C(X),O(C(X))〉 of X._With respect_to the natural order of_set-theoretical inclusion ⊆_of closed_subsets of X_the Vietoris space C(X) can be_conceived as an_order structure. Using the following elementary_reformulation_of the concept_of_a_pospace (cp._Gierz & others_1980,_p. 273)_one_can show that it is indeed_a_pospace: Theorem 5.2 Let 〈X,≤〉 be a_poset with topology O(X)._X_is a pospace if_& only if whenever a,_b ∈ X and NOT (a_≤ b),_there exist_open sets U, V ∈ O(X) with a ∈ U and_b ∈ V such that if_x ∈ U and_y ∈_V,_then NOT (x_≤_y). Theorem_5.3 Let X be a compact Hausdorff_space. Then_the Vietoris space C(X) of closed_subsets of X is_a_pospace. Proof : Assume a, b_∈ C(X) and NOT (a ≤_b). Hence there are α_∈_a_and β ∈ b. Since_X is normal, the closed sets_{α} and b_have disjoint open neighbourhoods V (α) and_R(b)._Let R(a) be an open set_containing_a. Note that a ∈ m(V_(α))._Hence_R(a) ∩ m(V (a)) is_an open neighbourhood