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Theorem List for Metamath Proof Explorer - 17801-17900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremqtopf1 17801 If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn              qTop

Theoremqtophmeo 17802* 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 homeomorhism that makes the diagram commute. (Contributed by Mario Carneiro, 24-Mar-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
TopOn                            qTop qTop

Theoremt0kq 17803* 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 17804 A topological space is T0 iff it is homeomorphic to its Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremist1-5lem 17805 Lemma for ist1-5 17807 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 17806 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 17807 A topological space is T1 iff it is both T0 and R0. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremishaus3 17808 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 17809 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 17734. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremreghaus 17810 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 17811 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.)

11.2  Filters and filter bases

11.2.1  Filter bases

Theoremelmptrab 17812* Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremelmptrab2 17813* Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremisfbas 17814* 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 17815 There are no empty filter bases. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theorem0nelfb 17816 No filter base contains the empty set. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremfbsspw 17817 A filter base on a set is a subset of the power set. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremfbelss 17818 An element of the filter base is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremfbdmn0 17819 The domain of a filter base is nonempty. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremisfbas2 17820* The predicate " is a filter base." (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfbasssin 17821* A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.)

Theoremfbssfi 17822* 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 17823* 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 17824 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 17825* 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 17826 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 17827* 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 17828 Conditions for the trace of a filter base to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
t t

Theoremtrfbas 17829* Conditions for the trace of a filter base to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
t

11.2.2  Filters

Syntaxcfil 17830 Extend class notation with the set of filters on a set.

Definitiondf-fil 17831* 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 17832* The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremfilfbas 17833 A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theorem0nelfil 17834 The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremfileln0 17835 An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremfilsspw 17836 A filter is a subset of the power set of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremfilelss 17837 An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremfilss 17838 A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfilin 17839 A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfiltop 17840 The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremisfil2 17841* Derive the standard axioms of a filter. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremisfildlem 17842* Lemma for isfild 17843. (Contributed by Mario Carneiro, 1-Dec-2013.)

Theoremisfild 17843* 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 17844 A filter is closed under taking intersections. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfilinn0 17845 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 17846 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 17847 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 17848 The intersection of two filters is a filter. Use fiint 7342 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 17849 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 17850 A filter base on any set is also a filter base on any larger set. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremsnfbas 17851 Condition for a singleton to be a filter base. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfsubbas 17852 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 17853 A filter base has the finite intersection property. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfbunfip 17854* 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 17855* 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 17856* A condition for elements of a generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremssfg 17857 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 17858 A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgss2 17859* 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 17860 A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremelfilss 17861* An element belongs to a filter iff any element below it does. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfilfinnfr 17862 No filter containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgcl 17863 A generated filter is a filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfgabs 17864 Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremneifil 17865 The neighborhoods of a non-empty 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 17866 Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfilunirn 17867 Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfilcon 17868 A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfbasrn 17869* 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 17870* 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 17871 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 17872* 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 17873 Conditions for the trace of a filter to be a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
t

Theoremtrfilss 17874 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 17875 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 17876 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 17875 for the converse cancellation law. (Contributed by Mario Carneiro, 15-Oct-2015.)
t

Theoremtrnei 17877 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 17872 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
TopOn t

Theoremcfinfil 17878* 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 17879* 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 17880* 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 17881 The set of upper integer sets is a filter base on , which corresponds to convergence of sequences on . (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremuzrest 17882 The restriction of the set of upper integers to an upper integer set is the set of upper integers based at a point above the cutoff. (Contributed by Mario Carneiro, 13-Oct-2015.)
t

Theoremuzfbas 17883 The set of upper integer sets 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.)

11.2.3  Ultrafilters

Syntaxcufil 17884 Extend class notation with the ultrafilters-on-a-set function.

Syntaxcufl 17885 Extend class notation with the ultrafilter lemma.
UFL

Definitiondf-ufil 17886* 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 17887* Define the class of base sets for which the ultrafilter lemma filssufil 17897 holds. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremisufil 17888* The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)

Theoremufilfil 17889 An ultrafilter is a filter. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)

Theoremufilss 17890 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 17891 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 17892 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.)

Theoremisufil2 17893* The maximal property of an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremufprim 17894 An ultrafilter is a prime filter. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Mario Carneiro, 2-Aug-2015.)

Theoremtrufil 17895 Conditions for the trace of an ultrafilter to be an ultrafilter. (Contributed by Mario Carneiro, 27-Aug-2015.)
t

Theoremfilssufilg 17896* A filter is contained in some ultrafilter. This version of filssufil 17897 contains the choice as a hypothesis (in the assumption that is well-orderable). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremfilssufil 17897* A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 8306.) (Contributed by Jeff Hankins, 2-Dec-2009.) (Revised by Stefan O'Rear, 29-Jul-2015.)

Theoremisufl 17898* Define the (strong) ultrafilter lemma, parameterized over base sets. A set satisfies the ultrafilter lemma if every filter on is a subset of some ultrafilter. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremufli 17899* Property of a set that satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

Theoremnumufl 17900 Consequence of filssufilg 17896: a set whose double powerset is well-orderable satifies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL

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