Huang Wenshan does form an equivalence relation. If formula_102 then formula_91 but the converse does not hold in general. However, if formula_89 is upward closed, such as a filter, then formula_91 if and only if formula_106 Every prefilter is equivalent to the filter that it generates. This shows that it is possible for filters to be equivalent to sets that are not filters. If two families of sets formula_85 and formula_89 are equivalent then either both formula_85 and formula_89 are ultra (resp. prefilters, filter subbases) or otherwise neither one of them is ultra (resp. a prefilter, a filter subbase). In particular, if 	Ultrafilter (set theory)
6822830	a filter subbase is not also a prefilter, then it is equivalent to the filter or prefilter that it generates. If formula_85 and formula_89 are both filters on formula_2 then formula_85 and formula_89 are equivalent if and only if formula_116 If a proper filter (resp. ultrafilter) is equivalent to a family of sets formula_85 then formula_85 is necessarily a prefilter (resp. ultra prefilter). Using the following characterization, it is possible to define prefilters (resp. ultra prefilters) using only the concept of filters (resp. ultrafilters) and subordination: Characterizations. There are no ultrafilters on the empty set, so it is henceforth assumed 	Ultrafilter (set theory)
6822831	that formula_2 is nonempty. A filter base formula_1 on formula_2 is an ultrafilter on formula_2 if and only if any of the following equivalent conditions hold: A (proper) filter formula_1 on formula_2 is an ultrafilter on formula_2 if and only if any of the following equivalent conditions hold: Grills and Filter-Grills. If formula_128 then its is the family formula_129 where formula_130 may be written if formula_2 is clear from context. For example, formula_132 and if formula_133 then formula_134 If formula_135 then formula_136 and moreover, if formula_137 is a filter subbase then formula_138 The grill formula_139 is upward closed in 	Ultrafilter (set theory)
6822832	formula_2 if and only if formula_141 which will henceforth be assumed. Moreover, formula_142 so that formula_137 is upward closed in formula_2 if and only if formula_145 The grill of a filter on formula_2 is called a For any formula_147 formula_137 is a filter-grill on formula_2 if and only if (1) formula_137 is upward closed in formula_2 and (2) for all sets formula_152 and formula_153 if formula_154 then formula_155 or formula_156 The grill operation formula_157 induces a bijection whose inverse is also given by formula_159 If formula_160 then formula_161 is a filter-grill on formula_2 if and only if formula_163 	Ultrafilter (set theory)
6822833	or equivalently, if and only if formula_161 is an ultrafilter on formula_165 That is, a filter on formula_2 is a filter-grill if and only if it is ultra. For any non-empty formula_167 formula_161 is both a filter on formula_2 and a filter-grill on formula_2 if and only if (1) formula_171 and (2) for all formula_172 the following equivalences hold: Free or principal. If formula_42 is any non-empty family of sets then the Kernel of formula_42 is the intersection of all sets in formula_178 formula_179 A non-empty family of sets formula_42 is called: If a family 	Ultrafilter (set theory)
6822834	of sets formula_42 is fixed then formula_42 is ultra if and only if some element of formula_42 is a singleton set, in which case formula_42 will necessarily be a prefilter. Every principal prefilter is fixed, so a principal prefilter formula_42 is ultra if and only if formula_185 is a singleton set. A singleton set is ultra if and only if its sole element is also a singleton set. Every filter on formula_2 that is principal at a single point is an ultrafilter, and if in addition formula_2 is finite, then there are no ultrafilters on formula_2 other than these. If 	Ultrafilter (set theory)
6822835	there exists a free ultrafilter (or even filter subbase) on a set formula_2 then formula_2 must be infinite. The next theorem shows that every ultrafilter falls into one of two categories: either it is free or else it is a principal filter generated by a single point. Examples, properties, and sufficient conditions. If formula_1 and formula_201 are families of sets such that formula_1 is ultra, formula_203 and formula_204 then formula_201 is necessarily ultra. A filter subbase formula_1 that is not a prefilter cannot be ultra; but it is nevertheless still possible for the prefilter and filter generated by formula_1 	Ultrafilter (set theory)
6822836	to be ultra. Suppose formula_120 is ultra and formula_209 is a set. The trace formula_210 is ultra if and only if it does not contain the empty set. Furthermore, at least one of the sets formula_211 and formula_212 will be ultra (this result extends to any finite partition of formula_2). If formula_214 are filters on formula_3 formula_1 is an ultrafilter on formula_3 and formula_218 then there is some formula_219 that satisfies formula_220 This result is not necessarily true for an infinite family of filters. The image under a map formula_221 of an ultra set formula_120 is again ultra and if 	Ultrafilter (set theory)
6822837	formula_1 is an ultra prefilter then so is formula_224 The property of being ultra is preserved under bijections. However, the preimage of an ultrafilter is not necessarily ultra, not even if the map is surjective. For example, if formula_2 has more than one point and if the range of formula_221 consists of a single point formula_227 then formula_227 is an ultra prefilter on formula_209 but its preimage is not ultra. Alternatively, if formula_1 is a principal filter generated by a point in formula_231then the preimage of formula_1 contains the empty set and so is not ultra. The elementary filter induced 	Ultrafilter (set theory)
6822838	by an infinite sequence, all of whose points are distinct, is an ultrafilter. If formula_233 then formula_234 denotes the set consisting all subsets of formula_2 having cardinality formula_236 and if formula_2contains at least formula_238 (formula_239) distinct points, then formula_234 is ultra but it is not contained in any prefilter. This example generalizes to any integer formula_241 and also to formula_242 if formula_2 contains more than one element. Ultra sets that are not also prefilters are rarely used. For every formula_244 and every formula_245 let formula_246 If formula_247 is an ultrafilter on formula_2 then the set of all formula_244 such that 	Ultrafilter (set theory)
6822839	formula_250 is an ultrafilter on formula_251 Monad structure. The functor associating to any set formula_2 the set of formula_253 of all ultrafilters on formula_2 forms a monad called the ultrafilter monad. The unit map sends any element formula_256 to the principal ultrafilter given by formula_188 This monad admits a conceptual explanation as the codensity monad of the inclusion of the category of finite sets into the category of all sets. The ultrafilter lemma. The ultrafilter lemma was first proved by Alfred Tarski in 1930. The ultrafilter lemma is equivalent to each of the following statements: A consequence of the ultrafilter 	Ultrafilter (set theory)
6822840	lemma is that every filter is equal to the intersection of all ultrafilters containing it. The following results can be proven using the ultrafilter lemma. A free ultrafilter exists on a set formula_2 if and only if formula_2 is infinite. Every proper filter is equal to the intersection of all ultrafilters containing it. Since there are filters that are not ultra, this shows that the intersection of a family of ultrafilters need not be ultra. A family of sets formula_264 can be extended to a free ultrafilter if and only if the intersection of any finite family of elements of 	Ultrafilter (set theory)
6822841	formula_265 is infinite. Relationships to other statements under ZF. Throughout this section, ZF refers to Zermelo–Fraenkel set theory and ZFC refers to ZF with the Axiom of Choice (AC). The ultrafilter lemma is independent of ZF. That is, there exist models in which the axioms of ZF hold but the ultrafilter lemma does not. There also exist models of ZF in which every ultrafilter is necessarily principal. Every filter that contains a singleton set is necessarily an ultrafilter and given formula_266 the definition of the discrete ultrafilter formula_267 does not require more than ZF. If formula_2 is finite then 	Ultrafilter (set theory)
6822842	every ultrafilter is a discrete filter at a point; consequently, free ultrafilters can only exist on infinite sets. In particular, if formula_2 is finite then the ultrafilter lemma can be proven from the axioms ZF. The existence of free ultrafilter on infinite sets can be proven if the axiom of choice is assumed. More generally, the ultrafilter lemma can be proven by using the axiom of choice, which in brief states that any Cartesian product of non-empty sets is non-empty. Under ZF, the axiom of choice is, in particular, equivalent to (a) Zorn's lemma, (b) Tychonoff's 	Ultrafilter (set theory)
6822843	theorem, (c) the weak form of the vector basis theorem (which states that every vector space has a basis), (d) the strong form of the vector basis theorem, and other statements. However, the ultrafilter lemma is strictly weaker than the axiom of choice. While free ultrafilters can be proven to exist, it is possible to construct an explicit example of a free ultrafilter; that is, free ultrafilters are intangible. Alfred Tarski proved that under ZFC, the cardinality of the set of all free ultrafilters on an infinite set formula_2 is equal to the cardinality of formula_271 where formula_4 denotes the 	Ultrafilter (set theory)
6822844	power set of formula_165 Other authors attribute this discovery to Bedřich Pospíšil (following a combinatorial argument from Fichtenholz, and Kantorovitch, improved by Hausdorff). Under ZF, the axiom of choice can be used to prove both the ultrafilter lemma and the Krein–Milman theorem; conversely, under ZF, the ultrafilter lemma together with the Krein–Milman theorem can prove the axiom of choice. Statements that cannot be deduced. The ultrafilter lemma is a relatively weak axiom. For example, each of the statements in the following list can be deduced from ZF together with the ultrafilter lemma: Equivalent statements. Under ZF, the 	Ultrafilter (set theory)
6822845	ultrafilter lemma is equivalent to each of the following statements: Completeness. The completeness of an ultrafilter formula_1 on a powerset is the smallest cardinal κ such that there are κ elements of formula_1 whose intersection is not in formula_20 The definition of an ultrafilter implies that the completeness of any powerset ultrafilter is at least formula_277. An ultrafilter whose completeness is than formula_277—that is, the intersection of any countable collection of elements of formula_1 is still in formula_1—is called countably complete or σ-complete. The completeness of a countably complete nonprincipal ultrafilter on a powerset is always a 	Ultrafilter (set theory)
6822846	measurable cardinal. The (named after Mary Ellen Rudin and Howard Jerome Keisler) is a preorder on the class of powerset ultrafilters defined as follows: if formula_1 is an ultrafilter on formula_82 and formula_283 an ultrafilter on formula_284 then formula_285 if there exists a function formula_221 such that for every subset formula_289 Ultrafilters formula_1 and formula_283 are called , denoted , if there exist sets formula_21 and formula_293 and a bijection formula_294 that satisfies the condition above. (If formula_2 and formula_209 have the same cardinality, the definition can be simplified by fixing formula_297 formula_298) It is known that ≡ is the kernel of 	Ultrafilter (set theory)
6822847	≤, i.e., that if and only if formula_299 and formula_300 Ultrafilters on formula_301. There are several special properties that an ultrafilter on formula_302 where formula_303 extends the natural numbers, may possess, which prove useful in various areas of set theory and topology. It is a trivial observation that all Ramsey ultrafilters are P-points. Walter Rudin proved that the continuum hypothesis implies the existence of Ramsey ultrafilters. In fact, many hypotheses imply the existence of Ramsey ultrafilters, including Martin's axiom. Saharon Shelah later showed that it is consistent that there are no P-point ultrafilters. Therefore, the existence of 	Ultrafilter (set theory)
6822848	these types of ultrafilters is independent of ZFC. P-points are called as such because they are topological P-points in the usual topology of the space of non-principal ultrafilters. The name Ramsey comes from Ramsey's theorem. To see why, one can prove that an ultrafilter is Ramsey if and only if for every 2-coloring of formula_320 there exists an element of the ultrafilter that has a homogeneous color. An ultrafilter on formula_301 is Ramsey if and only if it is minimal in the Rudin–Keisler ordering of non-principal powerset ultrafilters. Ultrafilter (set theory) In the 	Ultrafilter (set theory)
6822849	Huang Wenshan Huang Wenshan Huang Wenshan (黄文山, Pinyin: Huáng Wénshān, also known as Wen-shan Huang, 1898-1988) was a Chinese scholar of cultural studies, sociology, anthropology and ethnology. As a university student he was a well-known libertarian socialist during the May Fourth Movement. In his subsequent career he became a leading advocate for the science of culturology, lecturing and publishing on the topic in Chinese and English, at first under the pen name Huang Lingshuang (黃凌霜, Pinyin: Huáng Língshuāng) and later under his given name, while researching and teaching in China and the United States. He was also avid practitioner 	Huang Wenshan
6822850	and promoter of tai chi (taijiquan), and wrote one of the earliest comprehensive and popular books on that topic in English. Early years. Born in Taishan, Guangdong, Huang studied in the philosophy department of Peking University, where Zhu Qianzhi was a classmate. During the May Fourth Movement in 1919, Huang was elected editor-in-chief of the Peking University Student Weekly" by the Peking University Student Union. He and his successor in that post, fellow anarchist Zhu Qianzhi, organized a debate on anarchism there. In March 1920, Huang was cultivated as a contact by the first official representative of the  Communist International sent to China, Grigori Voitinsky, his wife Kuznetsova, their translator Yang Mingzhai and his entourage after their arrival in Beijing with the Soviet representative Stojanno, and later when they were active in Guangzhou. In 1921, Huang graduated from Peking University. In 1922, he went to study in the United States where he earned a master's degree at Columbia University. During his studies in the United States from 1922 to 1928, studied cultural theory under the anthropologist Alfred Louis Kluber. Academic career. In 1927, he served as a professor at National Labor University in Shanghai, and as a  professor and director of the Department of Social History of Jinan University. In 1930, Huang lived and translated books in Maojiazhuang, West Lake, and served as professor and director of the Department of Sociology at the National Central University in Nanjing. From the 1930s to the 1940s, he had contacts with scholars such as Yan Huanwen, Chen Gaozhu, Zhu Qianzhi, Cen Jiawu, and Chen Xujing. In the summer of 1936, Huang resigned as professor and director of the Department of Sociology of National Central University and director of ethnic studies at the Zhongshan Culture and Education Center. He returned to  Guangdong and became a professor at National Sun Yat-sen University in Guangzhou, where he founded programs for students of history, sociology, anthropology and other majors, and taught courses on culturology. In 1940, Huang served as professor and director of the Department of Sociology at National Sun Yat-sen University. In May 1941, he became the Dean of the Law School of National Sun Yat-sen University. In 1945, he served as the Dean of Guangdong Legislative Business School. In 1949, Huang left Mainland China for Taiwan. In 1950, at the invitation of Alfred Louis Kroeber Huang went to Columbia  University to serve as a guest scholar, and also taught at the New School in New York. He was given a grant from the Tsinghua Foundation (chaired by Mei Yiqi) to engage in cultural studies, and returned again to live in the United States. In 1960 he began teaching at the University of Southern California, in 1961 he served as dean of the Chinese Culture Institute in Los Angeles, and in 1962 he participated in the World Sociological Congress in Washington. Culturology. During the 1940s, Huang first began to correspond with American anthropologist Leslie White, who had coined the term  culturology in 1939 to describe the systematic study of culture and cultural systems. Huang went on teach university courses and publish widely on the topic, and became a recognized advocate of this approach. Russian-American sociologist Pitirim Sorokin in his 1966 book "Sociological Theories of Today" cited two of Huang's publications, "Collected Essays on Culturology" (Canton, 1939) and "Theoretical Trends of Culturology" (Taipei, 1959), as examples of how "Several scholars of various countries have advocated the establishment of a special science of culturology devoted to a study of cultural phenomena, differentiated from the science of sociology, which deals with  social phenomena." Huang's other publications on the topic included "Culturology: Its Evolution and Prospects" (Taipei, 1969), and "An Introduction to Culturology" (Hong Kong; Seattle, Wash, 1980) as well as a long list of Chinese-language books and journal articles in which the term culturology is translated as 文化学体系 (Pinyin: wénhuàxué tǐxì). Tai chi. An avid practitioner of tai chi (taijiquan) since his university days, in 1962 he founded the National Tai Chi Chuan Institute in Los Angeles, providing training space in the basement of a print shop he had established there. Marshall Ho'o assisted him in teaching the classes , which grew to hundreds of students in groups of thirty or more. Ho'o also helped him establish the National Tai Chi Chuan Association, linking schools throughout the United States. In 1967 after he had returned to Taiwan and turned leadership of the Institute and Association over to Ho'o, Huang sponsored tai chi master Tung Hu-ling on a teaching tour of North America including a term at the Institute in Los Angeles. In the 1930s and after World War II, Huang had trained under Tung Hu-ling's father Tung Ying-chieh. Ho'o, who regularly traveled to San Francisco's  Chinatown to seek out tai chi masters there, said Tung Hu-ling's visit was "the first time a tai chi master came to us". In 1973 Huang published the book "Fundamentals of Tai Chi Chuan", which included an introduction to Chinese culture and philosophy, and a detailed description of the art and its movements. One of the earliest comprehensive and popular books in English on the topic, it was subsequently released in three revised editions over next dozen years, adding an introduction by Laura Huxley and forewords by Justin F. Stone, Prestin K. Caye, and James C. Ingebretsen. Later  years. In 1967 Huang returned to Taiwan, where he continued to teach, lecture, research, and publish into the 1980s. He died in 1988. Huang Wenshan Huang Wenshan (黄文山, Pinyin: Huáng Wénshān, also known as Wen-shan Huang, 1898-1988) was a Chinese scholar of cultural studies, sociology, anthropology and ethnology. As a university student he was a well-known libertarian socialist during the May Fourth Movement. In his subsequent career he became a leading advocate for the science of culturology, lecturing and publishing on the topic in Chinese and English, at first under the pen name Huang Lingshuang (黃凌霜, Pinyin