## TPTP Axioms File: TOP001-0.ax

```%--------------------------------------------------------------------------
% File     : TOP001-0 : TPTP v7.5.0. Released v1.0.0.
% Domain   : Topology (Point set)
% Axioms   : Point-set topology
% Version  : [WM89] axioms : Incomplete.
% English  :

% Refs     : [WM89]  Wick & McCune (1989), Automated Reasoning about Elemen
% Source   : [WM89]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of clauses    :  109 (  23 non-Horn;   0 unit; 104 RR)
%            Number of atoms      :  336 (   0 equality)
%            Maximal clause size  :    8 (   3 average)
%            Number of predicates :   22 (   0 propositional; 1-4 arity)
%            Number of functors   :   35 (   1 constant; 0-5 arity)
%            Number of variables  :  357 (  56 singleton)
%            Maximal term depth   :    3 (   1 average)
% SPC      :

% Comments : These axioms are incomplete, in that they do not contain the
%            requisite set theory axioms. Problems that use this axiom set
%            must supply appropriate set theory axioms.
%--------------------------------------------------------------------------
%----Sigma (union of members).
cnf(union_of_members_1,axiom,
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) )).

cnf(union_of_members_2,axiom,
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) )).

cnf(union_of_members_3,axiom,
( element_of_set(U,union_of_members(Vf))
| ~ element_of_set(U,Uu1)
| ~ element_of_collection(Uu1,Vf) )).

%----Pi (intersection of members).
cnf(intersection_of_members_4,axiom,
( ~ element_of_set(U,intersection_of_members(Vf))
| ~ element_of_collection(Va,Vf)
| element_of_set(U,Va) )).

cnf(intersection_of_members_5,axiom,
( element_of_set(U,intersection_of_members(Vf))
| element_of_collection(f2(Vf,U),Vf) )).

cnf(intersection_of_members_6,axiom,
( element_of_set(U,intersection_of_members(Vf))
| ~ element_of_set(U,f2(Vf,U)) )).

%----Topological space
cnf(topological_space_7,axiom,
( ~ topological_space(X,Vt)
| equal_sets(union_of_members(Vt),X) )).

cnf(topological_space_8,axiom,
( ~ topological_space(X,Vt)
| element_of_collection(empty_set,Vt) )).

cnf(topological_space_9,axiom,
( ~ topological_space(X,Vt)
| element_of_collection(X,Vt) )).

cnf(topological_space_10,axiom,
( ~ topological_space(X,Vt)
| ~ element_of_collection(Y,Vt)
| ~ element_of_collection(Z,Vt)
| element_of_collection(intersection_of_sets(Y,Z),Vt) )).

cnf(topological_space_11,axiom,
( ~ topological_space(X,Vt)
| ~ subset_collections(Vf,Vt)
| element_of_collection(union_of_members(Vf),Vt) )).

cnf(topological_space_12,axiom,
( topological_space(X,Vt)
| ~ equal_sets(union_of_members(Vt),X)
| ~ element_of_collection(empty_set,Vt)
| ~ element_of_collection(X,Vt)
| element_of_collection(f3(X,Vt),Vt)
| subset_collections(f5(X,Vt),Vt) )).

cnf(topological_space_13,axiom,
( topological_space(X,Vt)
| ~ equal_sets(union_of_members(Vt),X)
| ~ element_of_collection(empty_set,Vt)
| ~ element_of_collection(X,Vt)
| element_of_collection(f3(X,Vt),Vt)
| ~ element_of_collection(union_of_members(f5(X,Vt)),Vt) )).

cnf(topological_space_14,axiom,
( topological_space(X,Vt)
| ~ equal_sets(union_of_members(Vt),X)
| ~ element_of_collection(empty_set,Vt)
| ~ element_of_collection(X,Vt)
| element_of_collection(f4(X,Vt),Vt)
| subset_collections(f5(X,Vt),Vt) )).

cnf(topological_space_15,axiom,
( topological_space(X,Vt)
| ~ equal_sets(union_of_members(Vt),X)
| ~ element_of_collection(empty_set,Vt)
| ~ element_of_collection(X,Vt)
| element_of_collection(f4(X,Vt),Vt)
| ~ element_of_collection(union_of_members(f5(X,Vt)),Vt) )).

cnf(topological_space_16,axiom,
( topological_space(X,Vt)
| ~ equal_sets(union_of_members(Vt),X)
| ~ element_of_collection(empty_set,Vt)
| ~ element_of_collection(X,Vt)
| ~ element_of_collection(intersection_of_sets(f3(X,Vt),f4(X,Vt)),Vt)
| subset_collections(f5(X,Vt),Vt) )).

cnf(topological_space_17,axiom,
( topological_space(X,Vt)
| ~ equal_sets(union_of_members(Vt),X)
| ~ element_of_collection(empty_set,Vt)
| ~ element_of_collection(X,Vt)
| ~ element_of_collection(intersection_of_sets(f3(X,Vt),f4(X,Vt)),Vt)
| ~ element_of_collection(union_of_members(f5(X,Vt)),Vt) )).

%----Open set
cnf(open_set_18,axiom,
( ~ open(U,X,Vt)
| topological_space(X,Vt) )).

cnf(open_set_19,axiom,
( ~ open(U,X,Vt)
| element_of_collection(U,Vt) )).

cnf(open_set_20,axiom,
( open(U,X,Vt)
| ~ topological_space(X,Vt)
| ~ element_of_collection(U,Vt) )).

%----Closed set
cnf(closed_set_21,axiom,
( ~ closed(U,X,Vt)
| topological_space(X,Vt) )).

cnf(closed_set_22,axiom,
( ~ closed(U,X,Vt)
| open(relative_complement_sets(U,X),X,Vt) )).

cnf(closed_set_23,axiom,
( closed(U,X,Vt)
| ~ topological_space(X,Vt)
| ~ open(relative_complement_sets(U,X),X,Vt) )).

%----Finer topology
cnf(finer_topology_24,axiom,
( ~ finer(Vt,Vs,X)
| topological_space(X,Vt) )).

cnf(finer_topology_25,axiom,
( ~ finer(Vt,Vs,X)
| topological_space(X,Vs) )).

cnf(finer_topology_26,axiom,
( ~ finer(Vt,Vs,X)
| subset_collections(Vs,Vt) )).

cnf(finer_topology_27,axiom,
( finer(Vt,Vs,X)
| ~ topological_space(X,Vt)
| ~ topological_space(X,Vs)
| ~ subset_collections(Vs,Vt) )).

%----Basis for a topology
cnf(basis_for_topology_28,axiom,
( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) )).

cnf(basis_for_topology_29,axiom,
( ~ basis(X,Vf)
| ~ element_of_set(Y,X)
| ~ element_of_collection(Vb1,Vf)
| ~ element_of_collection(Vb2,Vf)
| ~ element_of_set(Y,intersection_of_sets(Vb1,Vb2))
| element_of_set(Y,f6(X,Vf,Y,Vb1,Vb2)) )).

cnf(basis_for_topology_30,axiom,
( ~ basis(X,Vf)
| ~ element_of_set(Y,X)
| ~ element_of_collection(Vb1,Vf)
| ~ element_of_collection(Vb2,Vf)
| ~ element_of_set(Y,intersection_of_sets(Vb1,Vb2))
| element_of_collection(f6(X,Vf,Y,Vb1,Vb2),Vf) )).

cnf(basis_for_topology_31,axiom,
( ~ basis(X,Vf)
| ~ element_of_set(Y,X)
| ~ element_of_collection(Vb1,Vf)
| ~ element_of_collection(Vb2,Vf)
| ~ element_of_set(Y,intersection_of_sets(Vb1,Vb2))
| subset_sets(f6(X,Vf,Y,Vb1,Vb2),intersection_of_sets(Vb1,Vb2)) )).

cnf(basis_for_topology_32,axiom,
( basis(X,Vf)
| ~ equal_sets(union_of_members(Vf),X)
| element_of_set(f7(X,Vf),X) )).

cnf(basis_for_topology_33,axiom,
( basis(X,Vf)
| ~ equal_sets(union_of_members(Vf),X)
| element_of_collection(f8(X,Vf),Vf) )).

cnf(basis_for_topology_34,axiom,
( basis(X,Vf)
| ~ equal_sets(union_of_members(Vf),X)
| element_of_collection(f9(X,Vf),Vf) )).

cnf(basis_for_topology_35,axiom,
( basis(X,Vf)
| ~ equal_sets(union_of_members(Vf),X)
| element_of_set(f7(X,Vf),intersection_of_sets(f8(X,Vf),f9(X,Vf))) )).

cnf(basis_for_topology_36,axiom,
( basis(X,Vf)
| ~ equal_sets(union_of_members(Vf),X)
| ~ element_of_set(f7(X,Vf),Uu9)
| ~ element_of_collection(Uu9,Vf)
| ~ subset_sets(Uu9,intersection_of_sets(f8(X,Vf),f9(X,Vf))) )).

%----Topology generated by a basis
cnf(topology_generated_37,axiom,
( ~ element_of_collection(U,top_of_basis(Vf))
| ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X)) )).

cnf(topology_generated_38,axiom,
( ~ element_of_collection(U,top_of_basis(Vf))
| ~ element_of_set(X,U)
| element_of_collection(f10(Vf,U,X),Vf) )).

cnf(topology_generated_39,axiom,
( ~ element_of_collection(U,top_of_basis(Vf))
| ~ element_of_set(X,U)
| subset_sets(f10(Vf,U,X),U) )).

cnf(topology_generated_40,axiom,
( element_of_collection(U,top_of_basis(Vf))
| element_of_set(f11(Vf,U),U) )).

cnf(topology_generated_41,axiom,
( element_of_collection(U,top_of_basis(Vf))
| ~ element_of_set(f11(Vf,U),Uu11)
| ~ element_of_collection(Uu11,Vf)
| ~ subset_sets(Uu11,U) )).

%----Subspace topology
cnf(subspace_topology_42,axiom,
( ~ element_of_collection(U,subspace_topology(X,Vt,Y))
| topological_space(X,Vt) )).

cnf(subspace_topology_43,axiom,
( ~ element_of_collection(U,subspace_topology(X,Vt,Y))
| subset_sets(Y,X) )).

cnf(subspace_topology_44,axiom,
( ~ element_of_collection(U,subspace_topology(X,Vt,Y))
| element_of_collection(f12(X,Vt,Y,U),Vt) )).

cnf(subspace_topology_45,axiom,
( ~ element_of_collection(U,subspace_topology(X,Vt,Y))
| equal_sets(U,intersection_of_sets(Y,f12(X,Vt,Y,U))) )).

cnf(subspace_topology_46,axiom,
( element_of_collection(U,subspace_topology(X,Vt,Y))
| ~ topological_space(X,Vt)
| ~ subset_sets(Y,X)
| ~ element_of_collection(Uu12,Vt)
| ~ equal_sets(U,intersection_of_sets(Y,Uu12)) )).

%----Interior of a set
cnf(interior_47,axiom,
( ~ element_of_set(U,interior(Y,X,Vt))
| topological_space(X,Vt) )).

cnf(interior_48,axiom,
( ~ element_of_set(U,interior(Y,X,Vt))
| subset_sets(Y,X) )).

cnf(interior_49,axiom,
( ~ element_of_set(U,interior(Y,X,Vt))
| element_of_set(U,f13(Y,X,Vt,U)) )).

cnf(interior_50,axiom,
( ~ element_of_set(U,interior(Y,X,Vt))
| subset_sets(f13(Y,X,Vt,U),Y) )).

cnf(interior_51,axiom,
( ~ element_of_set(U,interior(Y,X,Vt))
| open(f13(Y,X,Vt,U),X,Vt) )).

cnf(interior_52,axiom,
( element_of_set(U,interior(Y,X,Vt))
| ~ topological_space(X,Vt)
| ~ subset_sets(Y,X)
| ~ element_of_set(U,Uu13)
| ~ subset_sets(Uu13,Y)
| ~ open(Uu13,X,Vt) )).

%----Closure of a set
cnf(closure_53,axiom,
( ~ element_of_set(U,closure(Y,X,Vt))
| topological_space(X,Vt) )).

cnf(closure_54,axiom,
( ~ element_of_set(U,closure(Y,X,Vt))
| subset_sets(Y,X) )).

cnf(closure_55,axiom,
( ~ element_of_set(U,closure(Y,X,Vt))
| ~ subset_sets(Y,V)
| ~ closed(V,X,Vt)
| element_of_set(U,V) )).

cnf(closure_56,axiom,
( element_of_set(U,closure(Y,X,Vt))
| ~ topological_space(X,Vt)
| ~ subset_sets(Y,X)
| subset_sets(Y,f14(Y,X,Vt,U)) )).

cnf(closure_57,axiom,
( element_of_set(U,closure(Y,X,Vt))
| ~ topological_space(X,Vt)
| ~ subset_sets(Y,X)
| closed(f14(Y,X,Vt,U),X,Vt) )).

cnf(closure_58,axiom,
( element_of_set(U,closure(Y,X,Vt))
| ~ topological_space(X,Vt)
| ~ subset_sets(Y,X)
| ~ element_of_set(U,f14(Y,X,Vt,U)) )).

%----Neighborhood
cnf(neighborhood_59,axiom,
( ~ neighborhood(U,Y,X,Vt)
| topological_space(X,Vt) )).

cnf(neighborhood_60,axiom,
( ~ neighborhood(U,Y,X,Vt)
| open(U,X,Vt) )).

cnf(neighborhood_61,axiom,
( ~ neighborhood(U,Y,X,Vt)
| element_of_set(Y,U) )).

cnf(neighborhood_62,axiom,
( neighborhood(U,Y,X,Vt)
| ~ topological_space(X,Vt)
| ~ open(U,X,Vt)
| ~ element_of_set(Y,U) )).

%----Limit point
cnf(limit_point_63,axiom,
( ~ limit_point(Z,Y,X,Vt)
| topological_space(X,Vt) )).

cnf(limit_point_64,axiom,
( ~ limit_point(Z,Y,X,Vt)
| subset_sets(Y,X) )).

cnf(limit_point_65,axiom,
( ~ limit_point(Z,Y,X,Vt)
| ~ neighborhood(U,Z,X,Vt)
| element_of_set(f15(Z,Y,X,Vt,U),intersection_of_sets(U,Y)) )).

cnf(limit_point_66,axiom,
( ~ limit_point(Z,Y,X,Vt)
| ~ neighborhood(U,Z,X,Vt)
| ~ eq_p(f15(Z,Y,X,Vt,U),Z) )).

cnf(limit_point_67,axiom,
( limit_point(Z,Y,X,Vt)
| ~ topological_space(X,Vt)
| ~ subset_sets(Y,X)
| neighborhood(f16(Z,Y,X,Vt),Z,X,Vt) )).

cnf(limit_point_68,axiom,
( limit_point(Z,Y,X,Vt)
| ~ topological_space(X,Vt)
| ~ subset_sets(Y,X)
| ~ element_of_set(Uu16,intersection_of_sets(f16(Z,Y,X,Vt),Y))
| eq_p(Uu16,Z) )).

%----Boundary of a set
cnf(boundary_69,axiom,
( ~ element_of_set(U,boundary(Y,X,Vt))
| topological_space(X,Vt) )).

cnf(boundary_70,axiom,
( ~ element_of_set(U,boundary(Y,X,Vt))
| element_of_set(U,closure(Y,X,Vt)) )).

cnf(boundary_71,axiom,
( ~ element_of_set(U,boundary(Y,X,Vt))
| element_of_set(U,closure(relative_complement_sets(Y,X),X,Vt)) )).

cnf(boundary_72,axiom,
( element_of_set(U,boundary(Y,X,Vt))
| ~ topological_space(X,Vt)
| ~ element_of_set(U,closure(Y,X,Vt))
| ~ element_of_set(U,closure(relative_complement_sets(Y,X),X,Vt)) )).

%----Hausdorff space
cnf(hausdorff_73,axiom,
( ~ hausdorff(X,Vt)
| topological_space(X,Vt) )).

cnf(hausdorff_74,axiom,
( ~ hausdorff(X,Vt)
| ~ element_of_set(X_1,X)
| ~ element_of_set(X_2,X)
| eq_p(X_1,X_2)
| neighborhood(f17(X,Vt,X_1,X_2),X_1,X,Vt) )).

cnf(hausdorff_75,axiom,
( ~ hausdorff(X,Vt)
| ~ element_of_set(X_1,X)
| ~ element_of_set(X_2,X)
| eq_p(X_1,X_2)
| neighborhood(f18(X,Vt,X_1,X_2),X_2,X,Vt) )).

cnf(hausdorff_76,axiom,
( ~ hausdorff(X,Vt)
| ~ element_of_set(X_1,X)
| ~ element_of_set(X_2,X)
| eq_p(X_1,X_2)
| disjoint_s(f17(X,Vt,X_1,X_2),f18(X,Vt,X_1,X_2)) )).

cnf(hausdorff_77,axiom,
( hausdorff(X,Vt)
| ~ topological_space(X,Vt)
| element_of_set(f19(X,Vt),X) )).

cnf(hausdorff_78,axiom,
( hausdorff(X,Vt)
| ~ topological_space(X,Vt)
| element_of_set(f20(X,Vt),X) )).

cnf(hausdorff_79,axiom,
( hausdorff(X,Vt)
| ~ topological_space(X,Vt)
| ~ eq_p(f19(X,Vt),f20(X,Vt)) )).

cnf(hausdorff_80,axiom,
( hausdorff(X,Vt)
| ~ topological_space(X,Vt)
| ~ neighborhood(Uu19,f19(X,Vt),X,Vt)
| ~ neighborhood(Uu20,f20(X,Vt),X,Vt)
| ~ disjoint_s(Uu19,Uu20) )).

%----Separation in a topological space
cnf(separation_81,axiom,
( ~ separation(Va1,Va2,X,Vt)
| topological_space(X,Vt) )).

cnf(separation_82,axiom,
( ~ separation(Va1,Va2,X,Vt)
| ~ equal_sets(Va1,empty_set) )).

cnf(separation_83,axiom,
( ~ separation(Va1,Va2,X,Vt)
| ~ equal_sets(Va2,empty_set) )).

cnf(separation_84,axiom,
( ~ separation(Va1,Va2,X,Vt)
| element_of_collection(Va1,Vt) )).

cnf(separation_85,axiom,
( ~ separation(Va1,Va2,X,Vt)
| element_of_collection(Va2,Vt) )).

cnf(separation_86,axiom,
( ~ separation(Va1,Va2,X,Vt)
| equal_sets(union_of_sets(Va1,Va2),X) )).

cnf(separation_87,axiom,
( ~ separation(Va1,Va2,X,Vt)
| disjoint_s(Va1,Va2) )).

cnf(separation_88,axiom,
( separation(Va1,Va2,X,Vt)
| ~ topological_space(X,Vt)
| equal_sets(Va1,empty_set)
| equal_sets(Va2,empty_set)
| ~ element_of_collection(Va1,Vt)
| ~ element_of_collection(Va2,Vt)
| ~ equal_sets(union_of_sets(Va1,Va2),X)
| ~ disjoint_s(Va1,Va2) )).

%----Connected topological space
cnf(connected_space_89,axiom,
( ~ connected_space(X,Vt)
| topological_space(X,Vt) )).

cnf(connected_space_90,axiom,
( ~ connected_space(X,Vt)
| ~ separation(Va1,Va2,X,Vt) )).

cnf(connected_space_91,axiom,
( connected_space(X,Vt)
| ~ topological_space(X,Vt)
| separation(f21(X,Vt),f22(X,Vt),X,Vt) )).

%----Connected set
cnf(connected_set_92,axiom,
( ~ connected_set(Va,X,Vt)
| topological_space(X,Vt) )).

cnf(connected_set_93,axiom,
( ~ connected_set(Va,X,Vt)
| subset_sets(Va,X) )).

cnf(connected_set_94,axiom,
( ~ connected_set(Va,X,Vt)
| connected_space(Va,subspace_topology(X,Vt,Va)) )).

cnf(connected_set_95,axiom,
( connected_set(Va,X,Vt)
| ~ topological_space(X,Vt)
| ~ subset_sets(Va,X)
| ~ connected_space(Va,subspace_topology(X,Vt,Va)) )).

%----Open covering
cnf(open_covering_96,axiom,
( ~ open_covering(Vf,X,Vt)
| topological_space(X,Vt) )).

cnf(open_covering_97,axiom,
( ~ open_covering(Vf,X,Vt)
| subset_collections(Vf,Vt) )).

cnf(open_covering_98,axiom,
( ~ open_covering(Vf,X,Vt)
| equal_sets(union_of_members(Vf),X) )).

cnf(open_covering_99,axiom,
( open_covering(Vf,X,Vt)
| ~ topological_space(X,Vt)
| ~ subset_collections(Vf,Vt)
| ~ equal_sets(union_of_members(Vf),X) )).

%----Compact topological space
cnf(compact_space_100,axiom,
( ~ compact_space(X,Vt)
| topological_space(X,Vt) )).

cnf(compact_space_101,axiom,
( ~ compact_space(X,Vt)
| ~ open_covering(Vf1,X,Vt)
| finite(f23(X,Vt,Vf1)) )).

cnf(compact_space_102,axiom,
( ~ compact_space(X,Vt)
| ~ open_covering(Vf1,X,Vt)
| subset_collections(f23(X,Vt,Vf1),Vf1) )).

cnf(compact_space_103,axiom,
( ~ compact_space(X,Vt)
| ~ open_covering(Vf1,X,Vt)
| open_covering(f23(X,Vt,Vf1),X,Vt) )).

cnf(compact_space_104,axiom,
( compact_space(X,Vt)
| ~ topological_space(X,Vt)
| open_covering(f24(X,Vt),X,Vt) )).

cnf(compact_space_105,axiom,
( compact_space(X,Vt)
| ~ topological_space(X,Vt)
| ~ finite(Uu24)
| ~ subset_collections(Uu24,f24(X,Vt))
| ~ open_covering(Uu24,X,Vt) )).

%----Compact set
cnf(compact_set_106,axiom,
( ~ compact_set(Va,X,Vt)
| topological_space(X,Vt) )).

cnf(compact_set_107,axiom,
( ~ compact_set(Va,X,Vt)
| subset_sets(Va,X) )).

cnf(compact_set_108,axiom,
( ~ compact_set(Va,X,Vt)
| compact_space(Va,subspace_topology(X,Vt,Va)) )).

cnf(compact_set_109,axiom,
( compact_set(Va,X,Vt)
| ~ topological_space(X,Vt)
| ~ subset_sets(Va,X)
| ~ compact_space(Va,subspace_topology(X,Vt,Va)) )).

%--------------------------------------------------------------------------
```