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Theorem List for Metamath Proof Explorer - 20901-21000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxpstopnlem1 20901* The function used in xpsval 15556 is a homeomorphism from the binary product topology to the indexed product topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
TopOn       TopOn

Theoremxpstps 20902 A binary product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremxpstopnlem2 20903* Lemma for xpstopn 20904. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremxpstopn 20904 The topology on a binary product of topological spaces, as we have defined it (transferring the indexed product topology on functions on to by the canonical bijection), coincides with the usual topological product (generated by a base of rectangles). (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremptcmpfi 20905 A topological product of finitely many compact spaces is compact. This weak version of Tychonoff's theorem does not require the axiom of choice. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theoremxkocnv 20906* The inverse of the "currying" function is the uncurrying function. (Contributed by Mario Carneiro, 13-Apr-2015.)
TopOn       TopOn              𝑛Locally        𝑛Locally

Theoremxkohmeo 20907* The Exponential Law for topological spaces. The "currying" function is a homeomorphism on function spaces when and are exponentiable spaces (by xkococn 20752, it is sufficient to assume that are locally compact to ensure exponentiability). (Contributed by Mario Carneiro, 13-Apr-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
TopOn       TopOn              𝑛Locally        𝑛Locally

Theoremqtopf1 20908 If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn              qTop

Theoremqtophmeo 20909* If two functions on a base topology make the same identifications in order to create quotient spaces qTop and qTop , then not only are qTop and qTop homeomorphic, but there is a unique homeomorphism that makes the diagram commute. (Contributed by Mario Carneiro, 24-Mar-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
TopOn                            qTop qTop

Theoremt0kq 20910* A topological space is T0 iff the quotient map is a homeomorphism onto the space's Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqhmph 20911 A topological space is T0 iff it is homeomorphic to its Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremist1-5lem 20912 Lemma for ist1-5 20914 and similar theorems. If is a topological property which implies T0, such as T1 or T2, the property can be "decomposed" into T0 and a non-T0 version of property (which is defined as stating that the Kolmogorov quotient of the space has property ). For example, if is T1, then the theorem states that a space is T1 iff it is T0 and its Kolmogorov quotient is T1 (we call this property R0). (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ KQ        KQ KQ        KQ

Theoremt1r0 20913 A T1 space is R0. That is, the Kolmogorov quotient of a T1 space is also T1 (because they are homeomorphic). (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremist1-5 20914 A topological space is T1 iff it is both T0 and R0. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremishaus3 20915 A topological space is Hausdorff iff it is both T0 and R1 (where R1 means that any two topologically distinct points are separated by neighborhoods). (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremnrmreg 20916 A normal T1 space is regular Hausdorff. In other words, a T4 space is T3 . One can get away with slightly weaker assumptions; see nrmr0reg 20841. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremreghaus 20917 A regular T0 space is Hausdorff. In other words, a T3 space is T2 . A regular Hausdorff or T0 space is also known as a T3 space. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremnrmhaus 20918 A T1 normal space is Hausdorff. A Hausdorff or T1 normal space is also known as a T4 space. (Contributed by Mario Carneiro, 24-Aug-2015.)

12.2  Filters and filter bases

12.2.1  Filter bases

Theoremelmptrab 20919* Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremelmptrab2OLD 20920* Obsolete version of elmptrab2 20921 as of 26-Mar-2021. (Contributed by Stefan O'Rear, 28-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremelmptrab2 20921* Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.) (Revised by AV, 26-Mar-2021.)

Theoremisfbas 20922* The predicate " is a filter base." Note that some authors require filter bases to be closed under pairwise intersections, but that is not necessary under our definition. One advantage of this definition is that tails in a directed set form a filter base under our meaning. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremfbasne0 20923 There are no empty filter bases. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theorem0nelfb 20924 No filter base contains the empty set. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremfbsspw 20925 A filter base on a set is a subset of the power set. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremfbelss 20926 An element of the filter base is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremfbdmn0 20927 The domain of a filter base is nonempty. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremisfbas2 20928* The predicate " is a filter base." (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfbasssin 20929* A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.)

Theoremfbssfi 20930* A filter base contains subsets of its finite intersections. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfbssint 20931* A filter base contains subsets of its finite intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfbncp 20932 A filter base does not contain complements of its elements. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfbun 20933* A necessary and sufficient condition for the union of two filter bases to also be a filter base. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfbfinnfr 20934 No filter base containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremopnfbas 20935* The collection of open supersets of a nonempty set in a topology is a neighborhoods of the set, one of the motivations for the filter concept. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)

Theoremtrfbas2 20936 Conditions for the trace of a filter base to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
t t

Theoremtrfbas 20937* Conditions for the trace of a filter base to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
t

12.2.2  Filters

Syntaxcfil 20938 Extend class notation with the set of filters on a set.

Definitiondf-fil 20939* The set of filters on a set. Definition 1 (axioms FI, FIIa, FIIb, FIII) of [BourbakiTop1] p. I.36. Filters are used to define the concept of limit in the general case. They are a generalization of the idea of neighborhoods. Suppose you are in . With neighborhoods you can express the idea of a variable that tends to a specific number but you can't express the idea of a variable that tends to infinity. Filters relax the "axioms" of neighborhoods and then succeed in expressing the idea of something that tends to infinity. Filters were invented by Cartan in 1937 and made famous by Bourbaki in his treatise. A notion similar to the notion of filter is the concept of net invented by Moore and Smith in 1922. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremisfil 20940* The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremfilfbas 20941 A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theorem0nelfil 20942 The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremfileln0 20943 An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremfilsspw 20944 A filter is a subset of the power set of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremfilelss 20945 An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremfilss 20946 A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfilin 20947 A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfiltop 20948 The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremisfil2 20949* Derive the standard axioms of a filter. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremisfildlem 20950* Lemma for isfild 20951. (Contributed by Mario Carneiro, 1-Dec-2013.)

Theoremisfild 20951* Sufficient condition for a set of the form to be a filter. (Contributed by Mario Carneiro, 1-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfilfi 20952 A filter is closed under taking intersections. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfilinn0 20953 The intersection of two elements of a filter can't be empty. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfilintn0 20954 A filter has the finite intersection property. Remark below definition 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 20-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfiln0 20955 The empty set is not a filter. Remark below def. 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 30-Oct-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoreminfil 20956 The intersection of two filters is a filter. Use fiint 7866 to extend this property to the intersection of a finite set of filters. Paragraph 3 of [BourbakiTop1] p. I.36. (Contributed by FL, 17-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremsnfil 20957 A singleton is a filter. Example 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfbasweak 20958 A filter base on any set is also a filter base on any larger set. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremsnfbas 20959 Condition for a singleton to be a filter base. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfsubbas 20960 A condition for a set to generate a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfbasfip 20961 A filter base has the finite intersection property. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfbunfip 20962* A helpful lemma for showing that certain sets generate filters. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgval 20963* The filter generating class gives a filter for every filter base. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremelfg 20964* A condition for elements of a generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremssfg 20965 A filter base is a subset of its generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgss 20966 A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgss2 20967* A condition for a filter to be finer than another involving their filter bases. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgfil 20968 A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremelfilss 20969* An element belongs to a filter iff any element below it does. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfilfinnfr 20970 No filter containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgcl 20971 A generated filter is a filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgabs 20972 Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremneifil 20973 The neighborhoods of a nonempty set is a filter. Example 2 of [BourbakiTop1] p. I.36. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
TopOn

Theoremfilunibas 20974 Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfilunirn 20975 Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfilcon 20976 A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfbasrn 20977* Given a filter on a domain, produce a filter on the range. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremfiluni 20978* The union of a nonempty set of filters with a common base and closed under pairwise union is a filter. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremtrfil1 20979 Conditions for the trace of a filter to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
t

Theoremtrfil2 20980* Conditions for the trace of a filter to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
t

Theoremtrfil3 20981 Conditions for the trace of a filter to be a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
t

Theoremtrfilss 20982 If is a member of the filter, then the filter truncated to is a subset of the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
t

Theoremfgtr 20983 If is a member of the filter, then truncating to and regenerating the behavior outside using recovers the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
t

Theoremtrfg 20984 The trace operation and the operation are inverses to one another in some sense, with growing the base set and ↾t shrinking it. See fgtr 20983 for the converse cancellation law. (Contributed by Mario Carneiro, 15-Oct-2015.)
t

Theoremtrnei 20985 The trace, over a set , of the filter of the neighborhoods of a point is a filter iff belongs to the closure of . (This is trfil2 20980 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
TopOn t

Theoremcfinfil 20986* Relative complements of the finite parts of an infinite set is a filter. When the set of the relative complements is called Frechet's filter and is used to define the concept of limit of a sequence. (Contributed by FL, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremcsdfil 20987* The set of all elements whose complement is dominated by the base set is a filter. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremsupfil 20988* The supersets of a nonempty set which are also subsets of a given base set form a filter. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)

Theoremzfbas 20989 The set of upper sets of integers is a filter base on , which corresponds to convergence of sequences on . (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremuzrest 20990 The restriction of the set of upper sets of integers to an upper set of integers is the set of upper sets of integers based at a point above the cutoff. (Contributed by Mario Carneiro, 13-Oct-2015.)
t

Theoremuzfbas 20991 The set of upper sets of integers based at a point in a fixed upper integer set like is a filter base on , which corresponds to convergence of sequences on . (Contributed by Mario Carneiro, 13-Oct-2015.)

12.2.3  Ultrafilters

Syntaxcufil 20992 Extend class notation with the ultrafilters-on-a-set function.

Syntaxcufl 20993 Extend class notation with the ultrafilter lemma.
UFL

Definitiondf-ufil 20994* Define the set of ultrafilters on a set. An ultrafilter is a filter that gives a definite result for every subset. (Contributed by Jeff Hankins, 30-Nov-2009.)

Definitiondf-ufl 20995* Define the class of base sets for which the ultrafilter lemma filssufil 21005 holds. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremisufil 20996* The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)

Theoremufilfil 20997 An ultrafilter is a filter. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)

Theoremufilss 20998 For any subset of the base set of an ultrafilter, either the set is in the ultrafilter or the complement is. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)

Theoremufilb 20999 The complement is in an ultrafilter iff the set is not. (Contributed by Mario Carneiro, 11-Dec-2013.) (Revised by Mario Carneiro, 29-Jul-2015.)

Theoremufilmax 21000 Any filter finer than an ultrafilter is actually equal to it. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)

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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41046
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