Example diagrams

Names and statements of the forms.

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0. .
1. : The Axiom of Choice: Every set of non-empty sets has a choice function.
2. Existence of successor cardinals: For every cardinal there is a cardinal such that and .
3. : For all infinite cardinals , .
4. Every infinite set is the union of some disjoint family of denumerable subsets. (Denumerable means .)
5. : Every denumerable set of non-empty denumerable subsets of has a choice function.
6. : The union of a denumerable family of denumerable subsets of is denumerable.
7. There is no infinite decreasing sequence of cardinals.
8. : Every denumerable family of non-empty sets has a choice function.
9. Finite Dedekind finite: (see Jech 1973b): (\ac{Howard/Yorke} \cite{1989}): Every Dedekind finite set is finite.
10. : Every denumerable family of non-empty finite sets has a choice function.
11. A Form of Restricted Choice for Families of Finite Sets: For every infinite set , has an infinite subset such that for every , , the set of all element subsets of has a choice function.
12. A Form of Restricted Choice for Families of Finite Sets: For every infinite set and every , there is an infinite subset of such the set of all element subsets of has a choice function.
13. Every Dedekind finite subset of is finite.
14. BPI: Every Boolean algebra has a prime ideal.
15. (KW), The Kinna-Wagner Selection Principle: For every set there is a function such that for all , if then .
16. : Every denumerable collection of non-empty sets each with power has a choice function.
17. Ramsey's Theorem I: If is an infinite set and the family of all 2 element subsets of is partitioned into 2 sets and , then there is an infinite subset such that all 2 element subsets of belong to or all 2 element subsets of belong to . (Also, see form 325.)
18. : The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.
19. A real function is analytically representable if and only if it is in Baire's classification.
20. If and are families of pairwise disjoint sets and for all , then .
21. If is well ordered, and are families of pairwise disjoint sets, and for all , then .
22. : If every member of an infinite set of cardinality has power , then the union has power .
23. : For every ordinal , if and every member of has cardinality , then .
24. : Every denumerable collection of non-empty sets each with power has a choice function.
25. is regular for all ordinals .
26. : The union of denumerably many sets each of power has power .
27. : The union of denumerably many sets each of power has power .
28. (Where is a prime) AL20(): Every vector space over has the property that every linearly independent subset can be extended to a basis. ( is the element field.)
29. If and and are families of pairwise disjoint sets and for all , then .
30. Ordering Principle: Every set can be linearly ordered.
31. : The countable union theorem: The union of a denumerable set of denumerable sets is denumerable.
32. : Every denumerable set of non-empty countable sets has a choice function.
33. If , : Every linearly ordered set of element sets has a choice function.
34. is regular.
35. The union of countably many meager subsets of is meager. (Meager sets are the same as sets of the first category.)
36. Compact T spaces are Loeb. (A space is Loeb if the set of non-empty closed sets has a choice function.)
37. Lebesgue measure is countably additive.
38. is not the union of a countable family of countable sets.
39. : Every set of non-empty sets such that has a choice function.
40. : Every well orderable set of non-empty sets has a choice function.
41. : For every cardinal , or .
42. Löwenheim-Skolem Theorem: If a countable family of first order sentences is satisfiable in a set then it is satisfiable in a countable subset of .
43. (DC), Principle of Dependent Choices: If is a relation on a non-empty set and then there is a sequence of elements of such that .
44. : Given a relation such that for every subset of a set with there is an with , then there is a function such that .
45. If , : Every set of -element sets has a choice function.
46. If is a finite subset of , : For every , every set of -element sets has a choice function.
47. If , : Every well ordered collection of -element sets has a choice function.
48. If is a finite subset of , : For every .
49. Order Extension Principle: Every partial ordering can be extended to a linear ordering.
50. Sikorski's Extension Theorem: Every homomorphism of a subalgebra of a Boolean algebra into a complete Boolean algebra can be extended to a homomorphism of into .
51. Cofinality Principle: Every linear ordering has a cofinal sub well ordering.
52. Hahn-Banach Theorem: If is a real vector space and satisfies and and is a subspace of and is linear and satisfies then can be extended to such that is linear and .
53. For all infinite cardinals , .
54. For all infinite cardinals , adj implies is an aleph. ( adj iff and .)
55. For all infinite cardinals and , if then, or .
56. . ( is Hartogs' aleph, the least not .)
57. If and are Dedekind finite sets then either or .
58. There is an ordinal such that . ( is Hartogs' aleph, the least not .)
59. If is a partial ordering that is not a well ordering, then there is no set such that (the usual injective cardinal ordering on ) is isomorphic to .
60. : Every set of non-empty, well orderable sets has a choice function.
61. ): For each , , every set of element sets has a choice function.
62. : Every set of non-empty finite sets has a choice function.
63. : Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.
64. (see Howard/Yorke 1989): There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.)
65. The Krein-Milman Theorem: Let be a compact convex set in a locally convex topological vector space . Then has an extreme point. (An extreme point is a point which is not an interior point of any line segment which lies in .)
66. Every vector space over a field has a basis.
67. (MC), The Axiom of Multiple Choice: For every set of non-empty sets there is a function such that and is finite).
68. Nielsen-Schreier Theorem: Every subgroup of a free group is free.
69. Every field has an algebraic closure.
70. There is a non-trivial ultrafilter on .
71. : or .
72. Artin-Schreier Theorem: Every field in which is not the sum of squares can be ordered. (The ordering, , must satisfy (a) for all and (b) and c.)
73. , : For every , if is an infinite family of element sets, then has an infinite subfamily with a choice function.
74. For every the following are equivalent: (1) is closed and bounded. \itemitem{(2)} Every sequence has a convergent subsequence with limit in A.
75. If a set has at least two elements, then it can be partitioned into well orderable subsets, each of which has at least two elements.
76. (-MC): For every family of pairwise disjoint non-empty sets, there is a function such that for each , f(x) is a non-empty countable subset of .
77. A linear ordering of a set is a well ordering if and only if has no infinite descending sequences.
78. Urysohn's Lemma: If and are disjoint closed sets in a normal space , then there is a continuous which is 1 everywhere in and 0 everywhere in .
79. can be well ordered.
80. : Every denumerable set of pairs has a choice function.
81. (For ) : For every set there is an ordinal and a one to one function . ( and . ( is equivalent to form 1 (AC) and is equivalent to the selection principle (form 15)).
82. (see Howard/Yorke 1989): If is infinite then is Dedekind infinite. ( is finite is Dedekind finite.)
83. (see Howard/Yorke 1989): -finite is equivalent to finite.
84. (see Howard/Yorke 1989): is -finite if and only if is Dedekind finite).
85. : Every family of denumerable sets has a choice function.
86. : If is a set of non-empty sets such that , then has a choice function.
87. : Given a relation such that for every subset of a set with , there is an with then there is a function such that () .
88. : Every family of pairs has a choice function.
89. Antichain Principle: Every partially ordered set has a maximal antichain.
90. : Every linearly ordered set can be well ordered.
91. : The power set of a well ordered set can be well ordered.
92. : Every well ordered family of non-empty subsets of has a choice function.
93. There is a non-measurable subset of .
94. : Every denumerable family of non-empty sets of reals has a choice function.
95. Existence of Complementary Subspaces over a Field : If is a field, then every vector space over has the property that if is a subspace of , then there is a subspace such that and generates .
96. Löwig's Theorem. If and are both bases for the vector space then .
97. Cardinal Representatives: For every set there is a function with domain such that for all , (i) and (ii) .
98. The set of all finite subsets of a Dedekind finite set is Dedekind finite.
99. Rado's Selection Lemma: Let be a family of finite subsets (of ) and suppose for each finite there is a function such that . Then there is an such that for every finite there is a finite such that and such that and agree on S.
100. Weak Partition Principle: For all sets and , if , then it is not the case that .
101. Partition Principle: If is a partition of , then .
102. For all Dedekind finite cardinals and , if then .
103. If is a linear ordering and then some initial segment of is uncountable.
104. There is a regular uncountable aleph.
105. There is a partially ordered set such that for no set is (the ordering on is the usual injective cardinal ordering) isomorphic to .
106. Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire.
107. M.~Hall's Theorem: Let be a collection of finite subsets (of a set ) then if for each finite there is an injective choice function on () then there is an injective choice function on . (That is, a 1-1 function such that .) (According to a theorem of P.~Hall () is equivalent to . P.~Hall's theorem does not require the axiom of choice.)
108. There is an ordinal such that is not the union of a denumerable set of denumerable sets.
109. Every field and every vector space over has the property that each linearly independent set can be extended to a basis.
110. Every vector space over has a basis.
111. : The union of an infinite well ordered set of 2-element sets is an infinite well ordered set.
112. : For every family of non-empty sets each of which can be linearly ordered there is a function such that for all , is a non-empty finite subset of .
113. Tychonoff's Compactness Theorem for Countably Many Spaces: The product of a countable set of compact spaces is compact.
114. Every A-bounded topological space is weakly Loeb. ( A-bounded means amorphous subsets are relatively compact. {\it Weakly Loeb} means the set of non-empty closed subsets has a multiple choice function.)
115. The product of weakly Loeb spaces is weakly Loeb.
116. Every compact space is weakly Loeb.
117. If is a measurable cardinal, then is the th inaccessible cardinal.
118. Every linearly orderable topological space is normal.
119. ,uniformly orderable with order type of the integers): Suppose is a set and there is a function such that for each is an ordering of of type (the usual ordering of the integers), then has a choice function.
120. If , : Every linearly ordered set of non-empty sets each of whose cardinality is in has a choice function.
121. : Every linearly ordered set of non-empty finite sets has a choice function.
122. : Every well ordered set of non-empty finite sets has a choice function.
123. : Uniform weak ultrafilter principle: For each family of infinite sets such that , is a non-principal ultrafilter on .
124. Every operator on a Hilbert space with an amorphous base is the direct sum of a finite matrix and a scalar operator. (A set is amorphous if it is not the union of two disjoint infinite sets.)
125. There does not exist an infinite, compact connected space. (A space is a space in which the intersection of any well orderable family of open sets is open.)
126. , Countable axiom of multiple choice: For every denumerable set of non-empty sets there is a function such that for all , is a non-empty finite subset of .
127. An amorphous power of a compact space, which as a set is well orderable, is well orderable.
128. Aczel's Realization Principle: On every infinite set there is a Hausdorff topology with an infinite set of non-isolated points.
129. For every infinite set , admits a partition into sets of order type . (For every infinite , there is a set such that is a partition of and for each , is an ordering of of type .)
130. is well orderable.
131. : For every denumerable family of pairwise disjoint non-empty sets, there is a function such that for each , f(x) is a non-empty countable subset of .
132. : Every infinite family of finite sets has an infinite subfamily with a choice function.
133. Every set is either well orderable or has an infinite amorphous subset.
134. If is an infinite space and is , then is countable. ( is ``hereditarily ''.)
135. If is a space with at least two points and is hereditarily metacompact then is countable. (A space is metacompact if every open cover has an open point finite refinement. If and are covers of a space , then is a {\it refinement} of if . is {\it point finite} if there are only finitely many such that .)
136. Surjective Cardinal Cancellation (depends on ): For all cardinals and , implies .
137. Suppose . If is a 1-1 map from into then there are partitions and of and such that maps onto .
138. Suppose . If is a partial map from onto (that is, the domain is a subset of ), then there are partitions and of and such that maps onto .
139. Using the discrete topology on 2, is compact.
140. Let be the set of all (undirected) infinite cycles of reals (Graphs whose vertices are real numbers, connected, no loops and each vertex adjacent to exactly two others). Then there is a function on such that for all , is a direction along .
141. [14 P()] with : Let be a collection of sets such that and suppose is a symmetric binary relation on such that for all finite there is an consistent choice function for . Then there is an consistent choice function for .
142. : There is a set of reals without the property of Baire.
143. : If is a connected relation ( or ) then contains a -maximal transitive subset.
144. Every set is almost well orderable.
145. Compact -spaces are Dedekind finite. (A -space is a topological space in which the intersection of a countable collection of open sets is open.)
146. : For every topological space , if is a continuous finite to one image of an A1 space then is an A1 space. ( is A1 means if covers then such that
147. : Every topological space can be covered by a well ordered family of discrete sets.
148. : For every topological space , if is well ordered, then has a well ordered base.
149. : Every topological space is a continuous, finite to one image of an A1 space.
150. : Every infinite set of denumerable sets has an infinite subset with a choice function.
151. (): The union of a well ordered set of denumerable sets is well orderable.
152. : Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets.
153. The closed unit ball of a Hilbert space is compact in the weak topology.
154. Tychonoff's Compactness Theorem for Countably Many Spaces: The product of countably many compact spaces is compact.
155. : There are no non-trivial Läuchli continua. (A Läuchli continuum is a strongly connected continuum. {\it Continuum} compact, connected, Hausdorff space; and {\it strongly connected} every continuous real valued function is constant.)
156. Theorem of Gelfand and Kolmogoroff: Two compact spaces are homeomorphic if their rings of real valued continuous functions are isomorphic.
157. Theorem of Goodner: A compact space is extremally disconnected (the closure of every open set is open) if and only if each non-empty subset of (set of continuous real valued functions on ) which is pointwise bounded has a supremum.
158. In every Hilbert space , if the closed unit ball is sequentially compact, then has an orthonormal basis.
159. The regular cardinals are cofinal in the class of ordinals.
160. No Dedekind finite set can be mapped onto an aleph.
161. Definability of cardinal addition in terms of : There is a first order formula whose only non-logical symbol is (for cardinals) that defines cardinal addition.
162. Non-existence of infinite units: There is no infinite cardinal number such that and for all cardinals and , or .
163. Every non-well-orderable set has an infinite, Dedekind finite subset.
164. Every non-well-orderable set has an infinite subset with a Dedekind finite power set.
165. : Every well ordered family of non-empty, well orderable sets has a choice function.
166. : Every infinite family of pairs has an infinite subfamily with a choice function.
167. , Partial Kinna-Wagner Principle: For every denumerable family such that for all , , there is an infinite subset and a function such that for all , .
168. Dual Cantor-Bernstein Theorem: and implies .
169. There is an uncountable subset of without a perfect subset.
170. .
171. If is a partial order such that is the denumerable union of finite sets and all antichains in are finite then for each denumerable family of dense sets there is a generic filter.
172. For every infinite set , if is hereditarily countable (that is, every is countable) then .
173. MPL: Metric spaces are para-Lindelöf.
174. : The representation theorem for multi-algebras with unary operations: Assume is a multi-algebra with unary operations (and no other operations). Then there is an algebra with unary operations and an equivalence relation on such that and are isomorphic multi-algebras.
175. Transitivity Condition: For all sets , there is a set abd a function such that is transitive and is a one to one function from onto . von
176. Every infinite, locally finite group has an infinite Abelian subgroup. ( Locally finite means every finite subset generates a finite subgroup.)
177. An infinite box product of regular spaces, each of cardinality greater than 1, is neither first countable nor connected.
178. If , and , , : If is any set of -element sets then there is a function with domain such that for all , and .
179. Suppose is an ordinal. , ).
180. Every Abelian group has a divisible hull. (If and are groups, is a divisible hull of means is a divisible group, is a subgroup of and for every non-zero , such that .)
181. : Every set of non-empty sets such that has a choice function.
182. There is an aleph whose cofinality is greater than .
183. There are no minimal sets. That is, there are no sets such that (1) is incomparable with \itemitem{(2)} for every and \itemitem{(3)} or .
184. Existence of a double uniformization: For all and , for all , if there is an infinite cardinal satisfying: (1) , and \itemitem{(2)} , , then such that for all such that and such that . ( is called a double uniformization of .)
185. Every linearly ordered Dedekind finite set is finite.
186. Every pair of cardinal numbers has a least upper bound (in the usual cardinal ordering.)
187. Every pair of cardinal numbers has a greatest lower bound (in the usual cardinal ordering.)
188. : For every Abelian group there is a projective Abelian group and a homomorphism from onto .
189. : For every Abelian group there is an injective Abelian group and a one to one homomorphism from into .
190. There is a non-trivial injective Abelian group.
191. : There is a set such that for every set , there is an ordinal and a function from onto .
192. sets: For every set there is a projective set and a function from onto .
193. : Every Abelian group is a homomorphic image of a free projective Abelian group.
194. or : If , has domain , and is in , then there is a sequence of elements of with for all .
195. Every general linear system has a linear global reaction.
196. and are not both measurable.
197. is the union of three sets with the property that for all there is a straight line such that .
198. For every set , if the only linearly orderable subsets of are the finite subsets of , then either is finite or has an amorphous subset.
199. (For ) If all , Dedekind finite subsets of are finite, then all Dedekind finite subsets of are finite.
200. For all infinite , .
201. Linking Axiom for Boolean Algebras: Every Boolean algebra has a maximal linked system. ( is linked if for all and .)
202. : Every linearly ordered family of non-empty sets has a choice function.
203. (disjoint,: Every partition of into non-empty subsets has a choice function.
204. For every infinite , there is a function from onto .
205. For all cardinals and , if and then there is a cardinal such that .
206. The existence of a non-principal ultrafilter: There exists an infinite set and a non-principal ultrafilter on .
207. : The union of sets each of cardinality has cardinality less than .
208. For all ordinals , .
209. There is an ordinal such that for all , if is a denumerable union of denumerable sets then cannot be partitioned into non-empty sets.
210. The commutator subgroup of a free group is free.
211. : Dependent choice for relations on : If satisfies then there is a sequence of real numbers such that .
212. : If is a relation on such that for all , there is a such that , then there is a function such that for all , .
213. : If then has a choice function.
214. : For every family of infinite sets, there is a function such that for all , is a non-empty subset of and .
215. If can be linearly ordered implies is finite), then is finite.
216. Every infinite tree has either an infinite chain or an infinite antichain.
217. Every infinite partially ordered set has either an infinite chain or an infinite antichain.
218. , relatively prime to ): , if is a set of non-empty sets, then there is a function such that for all , is a non-empty, finite subset of and is relatively prime to .
219. , relatively prime to ): For all non-zero , if is a set of non-empty well orderable sets, then there is a function such that for all , is a non-empty, finite subset of , and is relatively prime to .
220. Suppose and is a prime. Any two elementary Abelian -groups (all non-trivial elements have order ) of the same cardinality are isomorphic.
221. For all infinite , there is a non-principal measure on .
222. There is a non-principal measure on .
223. There is an infinite set and a non-principal measure on .
224. There is a partition of the real line into Borel sets such that for some , , . ( for is defined by induction, is an open subset of and for , if is even and if is odd.)
225. Every proper filter on can be extended to an ultrafilter.
226. Let be a commutative ring with identity, a proper subring containing 1 and a prime ideal in . Then there is a subring of and a prime ideal in such that (a) (b) (c) is multiplicatively closed and (d) if , then is multiplicatively closed.
227. For all groups , if every finite subgroup of can be fully ordered then can be fully ordered.
228. Every torsion free Abelian group can be fully ordered.
229. If is a partially ordered group, then can be extended to a linear order on if and only if for every finite set , with the identity for to , the signs () can be chosen so that (where is the normal sub-semi-group of generated by and where is the identity of .)
230. .
231. : The union of a well ordered collection of well orderable sets is well orderable.
232. Every metric space has a -point finite base.
233. If a field has an algebraic closure it is unique up to isomorphism.
234. There is a non-Ramsey set: There is a set of infinite subsets of such that for every infinite subset of , has a subset which is in and a subset which is not in .
235. If is a vector space and and are bases for then and are comparable.
236. If is a vector space with a basis and is a linearly independent subset of such that no proper extension of is a basis for , then is a basis for .
237. The order of any group is divisible by the order of any of its subgroups, (i.e., if is a subgroup of then there is a set such that .)
238. Every elementary Abelian group (that is, for some prime every non identity element has order ) is the direct sum of cyclic subgroups.
239. AL20(): Every vector space over has the property that every linearly independent subset of can be extended to a basis.
240. If a group satisfies ``every ascending chain of subgroups is finite," then every subgroup of is finitely generated.
241. Every algebraic closure of has a real closed subfield.
242. There is, up to an isomorphism, at most one algebraic closure of .
243. Every principal ideal domain is a unique factorization domain.
244. Every principal ideal domain has a maximal ideal.
245. There is a function such that for every , , is a function from onto .
246. The monadic theory theory of is recursive.
247. Every atomless Boolean algebra is Dedekind infinite.
248. For any , is the cardinal number of an infinite complete Boolean algebra if and only if .
249. If is an infinite tree in which every element has exactly 2 immediate successors then has an infinite branch.
250. : For every natural number , every well ordered family of element sets has a choice function.
251. The additive groups and are isomorphic.
252. The additive groups of and are isomorphic.
253. Łoś' Theorem: If is a relational system, any set and an ultrafilter in , then and are elementarily equivalent.
254. : Every directed relation in which ramified subsets have least upper bounds, has a maximal element.
255. : Every directed relation in which every ramified subset has an upper bound, has a maximal element.
256. : Every partially ordered set in which every forest has an upper bound, has a maximal element.
257. : Every transitive relation in which every partially ordered subset has an upper bound, has a maximal element.
258. : Every directed relation in which linearly ordered subsets have upper bounds, has a maximal element.
259. : If is a transitive and connected relation in which every well ordered subset has an upper bound, then has a maximal element.
260. : If is a transitive and connected relation in which every partially ordered subset has an upper bound, then has a maximal element.
261. : Every transitive relation in which every subset which is a tree has an upper bound, has a maximal element.
262. : Every transitive relation in which every ramified subset has an upper bound, has a maximal element.
263. : Every every relation which is antisymmetric and connected contains a -maximal partially ordered subset.
264. : Every connected relation contains a -maximal partially ordered set.
265. : Every relation contains a -maximal transitive subset.
266. : Every antisymmetric relation contains -maximal partially ordered subset.
267. There is no infinite, free complete Boolean algebra.
268. If is a lattice isomorphic to the lattice of subalgebras of some unary universal algebra (a unary universal algebra is one with only unary or nullary operations) and is an automorphism of of order 2 (that is, is the identity) then there is a unary algebra and an isomorphism from onto the lattice of subalgebras of with () for all .
269. For every cardinal , there is a set such that and there is a choice function on the collection of 2-element subsets of .
270. : The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs only in a finite number of formulas.
271. If , : The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs in at most formulas.
272. There is an such that neither nor has a perfect subset.
273. There is a subset of which is not Borel.
274. There is a cardinal number and an such that adj. (The expression `` adj means there are cardinals such that and and for all and if , then (Compare with [0 A]).
275. The sequence of cardinals has a unique minimal upper bound.
276. : For every set , is Dedekind finite if and only if or .
277. : Every non-well-orderable cardinal is decomposable.
278. In an integral domain , if every ideal is finitely generated then has a maximal proper ideal.
279. The Closed Graph Theorem for operations between Fréchet Spaces: Suppose and are Fréchet spaces, is linear and is closed in . Then is continuous.
280. There is a complete separable metric space with a subset which does not have the Baire property.
281. There is a Hilbert space and an unbounded linear operator on .
282. .
283. Cardinality of well ordered subsets: For all and for all infinite , where is the set of all well orderable subsets of .
284. A system of linear equations over a field has a solution in if and only if every finite sub-system has a solution in .
285. Let be a set and , then has a fixed point if and only if is not the union of three mutually disjoint sets , and such that for .
286. Extended Krein-Milman Theorem: Let K be a quasicompact (sometimes called convex-compact), convex subset of a locally convex topological vector space, then K has an extreme point.
287. The Hahn-Banach Theorem for Separable Normed Linear Spaces: Assume is a separable normed linear space and satisfies and and assume is a linear function from a subspace of into which satisfies , then can be extended to so that is linear and .
288. If , : Every denumerable set of -element sets has a choice function.
289. If is a set of subsets of a countable set and is closed under chain unions, then has a -maximal element.
290. For all infinite , .
291. For all infinite , .
292. : For each linearly ordered family of non-empty sets , there is a function such that for all is non-empty, finite subset of .
293. For all sets and , if can be linearly ordered and there is a mapping of onto , then can be linearly ordered.
294. Every linearly ordered -set is well orderable.
295. DO: Every infinite set has a dense linear ordering.
296. Part-: Every infinite set is the disjoint union of infinitely many infinite sets.
297. Extremally disconnected compact Hausdorff spaces are projective in the category of all compact Hausdorff spaces.
298. Every compact Hausdorff space has a Gleason cover.
299. Any extremally disconnected compact Hausdorff space is projective in the category of Boolean topological spaces.
300. Any continuous surjection between extremally disconnected compact Hausdorff spaces has an irreducible restriction to a closed subset of its domain.
301. Any continuous surjection between Boolean spaces has an irreducible restriction to a closed subset of its domain.
302. Any continuous surjection between compact Hausdorff spaces has an irreducible restriction to a closed subset of its domain.
303. If is a Boolean algebra, and is closed under , then there is a -maximal proper ideal of such that .
304. There does not exist a topological space such that every infinite subset of contains an infinite compact subset.
305. There are Vitali equivalence classes. ( Vitali equivalence classes are equivalence classes of the real numbers under the relation .).
306. The set of Vitali equivalence classes is linearly orderable. ( Vitali equivalence classes are equivalence classes of the real numbers under the relation .).
307. If is the cardinality of the set of Vitali equivalence classes, then , where is Hartogs aleph function and the Vitali equivalence classes are equivalence classes of the real numbers under the relation .
308. If is a prime and if is a set of finite groups, then the weak direct product has a maximal -subgroup.
309. The Banach-Tarski Paradox: There are three finite partitions , , and of such that is congruent to for and is congruent to for .
310. The Measure Extension Theorem: Suppose that is a subring (that is, and ) of a Boolean algebra and is a measure on (that is, , for , and .) then there is a measure on that extends .
311. Abelian groups are amenable. ( is amenable if there is a finitely additive measure on such that and , .)
312. A subgroup of an amenable group is amenable. ( is amenable if there is a finitely additive measure on such that and , .)
313. (the set of integers under addition) is amenable. ( is amenable if there is a finitely additive measure on such that and , .)
314. For every set and every permutation on there are two reflections and on such that and for every if then and . (A reflection is a permutation such that is the identity.)
315. , where
316. If a linearly ordered set has the fixed point property then is complete. ( has the fixed point property if every function satisfying has a fixed point, and ( is {\it complete} if every subset of has a least upper bound.)
317. Weak Sikorski Theorem: If is a complete, well orderable Boolean algebra and is a homomorphism of the Boolean algebra into where is a subalgebra of the Boolean algebra , then can be extended to a homomorphism of into .
318. is not measurable.
319. Measurable cardinals are inaccessible.
320. No successor cardinal, , is measurable.
321. There does not exist an ordinal such that is weakly compact and is measurable.
322. , The Kinna-Wagner Selection Principle for a well ordered family of sets: For every well ordered set there is a function such that for all , if then .
323. , The Kinna-Wagner Selection Principle for a family of well orderable sets: For every set of well orderable sets there is a function such that for all , if then .
324. , The Kinna-Wagner Selection Principle for a well ordered family of well orderable sets: For every well ordered set of well orderable sets, there is a function such that for all , if then .
325. Ramsey's Theorem II: , if A is an infinite set and the family of all element subsets of is partitioned into sets , then there is an infinite subset such that all element subsets of belong to the same . (Also, see form 17.)
326. 2-SAT: Restricted Compactness Theorem for Propositional Logic III: If is a set of formulas in a propositional language such that every finite subset of is satisfiable and if every formula in is a disjunction of at most two literals, then is satisfiable. (A literal is a propositional variable or its negation.)
327. , The Kinna-Wagner Selection Principle for a well ordered family of finite sets: For every well ordered set of finite sets there is a function such that for all , if then .
328. : For every well ordered set such that for all , , there is a function such that and for every , is a finite, non-empty subset of .
329. : For every set of well orderable sets such that for all , , there is a function such that for every , is a finite, non-empty subset of .
330. : For every well ordered set of well orderable sets such that for all , , there is a function such that for every , is a finite, non-empty subset of .
331. If is a family of compact non-empty topological spaces then there is a family such that , is an irreducible closed subset of .
332. A product of non-empty compact sober topological spaces is non-empty.
333. : For every set of sets such that for all , , there is a function such that for every , is a finite, non-empty subset of and is odd.
334. : For every set of sets such that for all , , there is a function such that for every , is a finite, non-empty subset of and is even.
335. Every quotient group of an Abelian group each of whose elements has order has a set of representatives.
336. (For , .) For every infinite set , there is an infinite such that the set of all -element subsets of has a choice function.
337. : If is a well ordered collection of non-empty sets and there is a function defined on such that for every , is a linear ordering of , then there is a choice function for .
338. : The union of a denumerable number of denumerable sets is well orderable.
339. Martin's Axiom : Whenever is a non-empty, ccc quasi-order (ccc means every anti-chain is countable) and is a family of dense subsets of , then there is a generic filter in .
340. Every Lindelöf metric space is separable.
341. Every Lindelöf metric space is second countable.
342. (For , .) : Every infinite family of -element sets has an infinite subfamily with a choice function.
343. A product of non-empty, compact topological spaces is non-empty.
344. If is a family of non-empty sets, then there is a family such that , is an ultrafilter on .
345. Rasiowa-Sikorski Axiom: If is a Boolean algebra, is a non-zero element of , and is a denumerable set of subsets of then there is a maximal filter of such that and for each , if and exists then .
346. If is a vector space without a finite basis then contains an infinite, well ordered, linearly independent subset.
347. Idemmultiple Partition Principle: If is idemmultiple () and , then .
348. If is a group and and both freely generate then .
349. : For every set of non-empty denumerable sets there is a function such that for all , is a finite, non-empty subset of .
350. : For every denumerable set of non-empty denumerable sets there is a function such that for all , is a finite, non-empty subset of .
351. A countable product of metrizable spaces is metrizable.
352. A countable product of second countable spaces is second countable.
353. A countable product of first countable spaces is first countable.
354. A countable product of separable spaces is separable.
355. , The Kinna-Wagner Selection Principle for a denumerable family of sets: For every denumerable set there is a function such that for all , if then .
356. , The Kinna-Wagner Selection Principle for a family of denumerable sets: For every set of denumerable sets there is a function such that for all , if then .
357. , The Kinna-Wagner Selection Principle for a denumerable family of denumerable sets: For every denumerable set of denumerable sets there is a function such that for all , if then .
358. , The Kinna-Wagner Selection Principle for a denumerable family of finite sets: For every denumerable set of finite sets there is a function such that for all , if then .
359. If and are families of pairwise disjoint sets and for all , then .
361. In , the union of a denumerable number of analytic sets is analytic.
362. In , every Borel set is analytic.
363. There are exactly Borel sets in .
364. In , there is a measurable set that is not Borel.
365. For every uncountable set , if has the same cardinality as each of its uncountable subsets then .
366. There is a discontinuous function such that for all real and , .
367. There is a Hamel basis for as a vector space over .
368. The set of all denumerable subsets of has power .
369. If is partitioned into two sets, at least one of them has cardinality .
370. Weak Gelfand Extreme Point Theorem: If is a non-trivial Gelfand algebra then the closed unit ball in the dual of has an extreme point .
371. There is an infinite, compact, Hausdorff, extremally disconnected topological space.
372. Generalized Hahn-Banach Theorem: Assume that is a real vector space, is a Dedekind complete ordered vector space and is a subspace of . If is linear and is sublinear and if on then can be extended to a linear map such that on .
373. (For , .) : Every denumerable set of -element sets has an infinite subset with a choice function.
375. Tietze-Urysohn Extension Theorem: If is a normal topological space, is closed in , and is continuous, then there exists a continuous function which extends .
376. Restricted Kinna Wagner Principle: For every infinite set there is an infinite subset of and a function such that for every , if then is a non-empty proper subset of . De la Cruz/Di
377. Restricted Ordering Principle: For every infinite set there is an infinite subset of such that can be linearly ordered. De la Cruz/Di
378. Restricted Choice for Families of Well Ordered Sets: For every infinite set there is an infinite subset of such that the family of non-empty well orderable subsets of has a choice function. De la Cruz/Di
379. : For every infinite family of sets each of which has at least two elements, there is an infinite subfamily of and a function such that for all , is a non-empty proper subset of . De la Cruz/Di
380. : For every infinite family of non-empty well orderable sets, there is an infinite subfamily of which has a choice function. De la Cruz/Di
381. DUM: The disjoint union of metrizable spaces is metrizable.
382. DUMN: The disjoint union of metrizable spaces is normal.
384. Closed Filter Extendability for Spaces. Every closed filter in a topological space can be extended to a maximal closed filter.
385. Countable Ultrafilter Theorem: Every proper filter with a countable base over a set (in ) can be extended to an ultrafilter.
386. Every B compact (pseudo)metric space is Baire.
387. DPO: Every infinite set has a non-trivial, dense partial order. (A partial ordering on a set is dense if and is non-trivial if ).
388. Every infinite branching poset (a partially ordered set in which each element has at least two lower bounds) has either an infinite chain or an infinite antichain.
389. : Every denumerable family of two element subsets of has a choice function.
390. Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements.
391. : Every set of non-empty linearly orderable sets has a choice function.
392. : Every linearly ordered set of linearly orderable sets has a choice function.
393. : Every linearly ordered set of non-empty well orderable sets has a choice function.
394. : Every well ordered set of non-empty linearly orderable sets has a choice function.
395. : For each linearly ordered family of non-empty linearly orderable sets , there is a function such that for all is a non-empty, finite subset of .
396. : For each linearly ordered family of non-empty well orderable sets , there is a function such that for all is a non-empty, finite subset of .
397. : For each well ordered family of non-empty linearly orderable sets , there is a function such that for all is a non-empty, finite subset of .
398. , The Kinna-Wagner Selection Principle for a linearly ordered family of sets: For every linearly ordered set there is a function such that for all , if then .
399. , The Kinna-Wagner Selection Principle for a set of linearly orderable sets: For every set of linearly orderable sets there is a function such that for all , if then .
400. , The Kinna-Wagner Selection Principle for a linearly ordered set of linearly orderable sets: For every linearly ordered set of linearly orderable sets there is a function such that for all , if then .
401. , The Kinna-Wagner Selection Principle for a linearly ordered set of finite sets: For every linearly ordered set of finite sets there is a function such that for all , if then .
402. , The Kinna-Wagner Selection Principle for a well ordered set of linearly orderable sets: For every well ordered set of linearly orderable sets there is a function such that for all , if then .
403. , The Kinna-Wagner Selection Principle for a linearly ordered set of well orderable sets: For every linearly ordered set of well orderable sets there is a function such that for all , if then .
404. Every infinite set can be partitioned into infinitely many sets, each of which has at least two elements.
405. Every infinite set can be partitioned into sets each of which is countable and has at least two elements.
406. The product of compact Hausdorf spaces is countably compact.
407. Let be a Boolean algebra, a non-zero element of and a sequence of subsets of such that for each , has a supremum . Then there exists an ultrafilter in such that and, for each , if , then .
408. If is a family of functions such that for each , , where and are non-empty sets, and is a filter base on such that 1. For all and all finite there is an such that is defined on , and \itemitem{2.} For all and all finite there exist at most finitely many functions on which are restrictions of the functions with , \noindent then there is a function with domain such that for each finite and each there is an such that .
409. Suppose is a locally finite graph (i.e. is a non-empty set and is a function from to such that for each , and are finite), is a finite set of integers, and is a function mapping subsets of into subsets of . If for each finite subgraph there is a function such that for each , , then there is a function such that for all , .
410. RC (Reflexive Compactness): The closed unit ball of a reflexive normed space is compact for the weak topology.
411. RCuc (Reflexive Compactness for uniformly convex Banach spaces): The closed unit ball of a uniformly convex Banach space is compact for the weak topology.
412. RCh (Reflexive Compactness for Hilbert spaces): The closed unit ball of a Hilbert space is compact for the weak topology.
413. Every infinite set is the union of a set, well-ordered by inclusion, of subsets which are non-equipollent to .
414. Every -frame is a -frame.
415. Every -compactly generated complete lattice is algebraic.
416. Every non-compact topological space is the union of a set that is well-ordered by inclusion and consists of open proper subsets of .
417. On every non-trivial Banach space there is a non-trivial linear functional (bounded or unbounded).
418. DUM(): The countable disjoint union of metrizable spaces is metrizable.
419. UT(,cuf,cuf): The union of a denumerable set of cuf sets is cuf. (A set is cuf if it is a countable union of finite sets.)
420. UT(,,cuf): The union of a denumerable set of denumerable sets is cuf.
421. : The union of a denumerable set of well orderable sets can be well ordered.
422. , : The union of a well ordered set of element sets can be well ordered.
424. Every Lindelöf metric space is super second countable.
425. For every first countable topological space there is a family such that , countable local base at .
426. If is a first countable topological space and is a family such that for all , is a local base at , then there is a family such that for every , is a countable local base at and .
427. AL20(): There is a field such that every vector space over has the property that every independent subset of can be extended to a basis.
428. B: There is a field such that every vector space over has a basis.
429. (Where is a prime) B: Every vector space over has a basis. ( is the element field.)
430. (Where is a prime) AL21: Every vector space over has the property that for every subspace of , there is a subspace of such that and generates in other words such that .