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Type | Label | Description |
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Statement | ||
Theorem | hmeoqtop 20801 | A homeomorphism is a quotient map. (Contributed by Mario Carneiro, 25-Aug-2015.) |
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Theorem | hmph 20802 |
Express the predicate ![]() ![]() |
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Theorem | hmphi 20803 | If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.) |
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Theorem | hmphtop 20804 | Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.) |
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Theorem | hmphtop1 20805 | The relation "being homeomorphic to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
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Theorem | hmphtop2 20806 | The relation "being homeomorphic to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
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Theorem | hmphref 20807 | "Is homeomorphic to" is reflexive. (Contributed by FL, 25-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
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Theorem | hmphsym 20808 | "Is homeomorphic to" is symmetric. (Contributed by FL, 8-Mar-2007.) (Proof shortened by Mario Carneiro, 30-May-2014.) |
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Theorem | hmphtr 20809 | "Is homeomorphic to" is transitive. (Contributed by FL, 9-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
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Theorem | hmpher 20810 | "Is homeomorphic to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
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Theorem | hmphen 20811 | Homeomorphisms preserve the cardinality of the topologies. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
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Theorem | hmphsymb 20812 | "Is homeomorphic to" is symmetric. (Contributed by FL, 22-Feb-2007.) |
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Theorem | haushmphlem 20813* |
Lemma for haushmph 20818 and similar theorems. If the topological
property
![]() ![]() |
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Theorem | cmphmph 20814 | Compactness is a topological property-that is, for any two homeomorphic topologies, either both are compact or neither is. (Contributed by Jeff Hankins, 30-Jun-2009.) (Revised by Mario Carneiro, 23-Aug-2015.) |
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Theorem | conhmph 20815 | Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.) |
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Theorem | t0hmph 20816 | T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
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Theorem | t1hmph 20817 | T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
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Theorem | haushmph 20818 | Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
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Theorem | reghmph 20819 | Regularity is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
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Theorem | nrmhmph 20820 | Normality is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
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Theorem | hmph0 20821 | A topology homeomorphic to the empty set is empty. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
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Theorem | hmphdis 20822 | Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.) |
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Theorem | hmphindis 20823 | Homeomorphisms preserve topological indiscretion. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
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Theorem | indishmph 20824 | Equinumerous sets equipped with their indiscrete topologies are homeomorphic (which means in that particular case that a segment is homeomorphic to a circle contrary to what Wikipedia claims). (Contributed by FL, 17-Aug-2008.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) |
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Theorem | hmphen2 20825 | Homeomorphisms preserve the cardinality of the underlying sets. (Contributed by FL, 17-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
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Theorem | cmphaushmeo 20826 | A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. (Contributed by Mario Carneiro, 17-Feb-2015.) |
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Theorem | ordthmeolem 20827 | Lemma for ordthmeo 20828. (Contributed by Mario Carneiro, 9-Sep-2015.) |
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Theorem | ordthmeo 20828 | An order isomorphism is a homeomorphism on the respective order topologies. (Contributed by Mario Carneiro, 9-Sep-2015.) |
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Theorem | txhmeo 20829* | Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.) |
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Theorem | txswaphmeolem 20830* | Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.) |
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Theorem | txswaphmeo 20831* |
There is a homeomorphism from ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | pt1hmeo 20832* | The canonical homeomorphism from a topological product on a singleton to the topology of the factor. (Contributed by Mario Carneiro, 3-Feb-2015.) |
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Theorem | ptuncnv 20833* |
Exhibit the converse function of the map ![]() |
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Theorem | ptunhmeo 20834* |
Define a homeomorphism from a binary product of indexed product
topologies to an indexed product topology on the union of the index
sets. This is the topological analogue of
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | xpstopnlem1 20835* |
The function ![]() |
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Theorem | xpstps 20836 | A binary product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.) |
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Theorem | xpstopnlem2 20837* | Lemma for xpstopn 20838. (Contributed by Mario Carneiro, 27-Aug-2015.) |
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Theorem | xpstopn 20838 |
The topology on a binary product of topological spaces, as we have
defined it (transferring the indexed product topology on functions on
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ptcmpfi 20839 | 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.) |
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Theorem | xkocnv 20840* |
The inverse of the "currying" function ![]() |
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Theorem | xkohmeo 20841* |
The Exponential Law for topological spaces. The "currying" function
![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | qtopf1 20842 | If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.) |
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Theorem | qtophmeo 20843* |
If two functions on a base topology ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | t0kq 20844* | 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.) |
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Theorem | kqhmph 20845 | A topological space is T0 iff it is homeomorphic to its Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.) |
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Theorem | ist1-5lem 20846 |
Lemma for ist1-5 20848 and similar theorems. If ![]() ![]() ![]() ![]() |
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Theorem | t1r0 20847 | 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.) |
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Theorem | ist1-5 20848 | A topological space is T1 iff it is both T0 and R0. (Contributed by Mario Carneiro, 25-Aug-2015.) |
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Theorem | ishaus3 20849 | 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.) |
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Theorem | nrmreg 20850 | 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 20775. (Contributed by Mario Carneiro, 25-Aug-2015.) |
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Theorem | reghaus 20851 | 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.) |
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Theorem | nrmhaus 20852 | 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.) |
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Theorem | elmptrab 20853* | Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.) |
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Theorem | elmptrab2 20854* | Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.) |
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Theorem | isfbas 20855* |
The predicate "![]() |
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Theorem | fbasne0 20856 | There are no empty filter bases. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.) |
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Theorem | 0nelfb 20857 | No filter base contains the empty set. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.) |
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Theorem | fbsspw 20858 | A filter base on a set is a subset of the power set. (Contributed by Stefan O'Rear, 28-Jul-2015.) |
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Theorem | fbelss 20859 | An element of the filter base is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.) |
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Theorem | fbdmn0 20860 | The domain of a filter base is nonempty. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
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Theorem | isfbas2 20861* |
The predicate "![]() |
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Theorem | fbasssin 20862* | A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.) |
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Theorem | fbssfi 20863* | A filter base contains subsets of its finite intersections. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
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Theorem | fbssint 20864* | A filter base contains subsets of its finite intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
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Theorem | fbncp 20865 | 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.) |
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Theorem | fbun 20866* | 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.) |
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Theorem | fbfinnfr 20867 | No filter base containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
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Theorem | opnfbas 20868* | 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.) |
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Theorem | trfbas2 20869 |
Conditions for the trace of a filter base ![]() |
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Theorem | trfbas 20870* |
Conditions for the trace of a filter base ![]() |
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Syntax | cfil 20871 | Extend class notation with the set of filters on a set. |
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Definition | df-fil 20872* |
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 ![]() |
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Theorem | isfil 20873* | The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.) |
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Theorem | filfbas 20874 | A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.) |
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Theorem | 0nelfil 20875 | The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.) |
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Theorem | fileln0 20876 | An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.) |
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Theorem | filsspw 20877 | A filter is a subset of the power set of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.) |
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Theorem | filelss 20878 | An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.) |
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Theorem | filss 20879 | A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
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Theorem | filin 20880 | A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
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Theorem | filtop 20881 | The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
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Theorem | isfil2 20882* | Derive the standard axioms of a filter. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
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Theorem | isfildlem 20883* | Lemma for isfild 20884. (Contributed by Mario Carneiro, 1-Dec-2013.) |
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Theorem | isfild 20884* |
Sufficient condition for a set of the form ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | filfi 20885 | A filter is closed under taking intersections. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
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Theorem | filinn0 20886 | 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.) |
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Theorem | filintn0 20887 | 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.) |
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Theorem | filn0 20888 | 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.) |
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Theorem | infil 20889 | The intersection of two filters is a filter. Use fiint 7835 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.) |
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Theorem | snfil 20890 | 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.) |
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Theorem | fbasweak 20891 | A filter base on any set is also a filter base on any larger set. (Contributed by Stefan O'Rear, 2-Aug-2015.) |
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Theorem | snfbas 20892 | Condition for a singleton to be a filter base. (Contributed by Stefan O'Rear, 2-Aug-2015.) |
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Theorem | fsubbas 20893 | 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.) |
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Theorem | fbasfip 20894 | A filter base has the finite intersection property. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
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Theorem | fbunfip 20895* | 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.) |
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Theorem | fgval 20896* | 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.) |
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Theorem | elfg 20897* | A condition for elements of a generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
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Theorem | ssfg 20898 | 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.) |
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Theorem | fgss 20899 | A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
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Theorem | fgss2 20900* | 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.) |
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