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

Theoremhmeontr 17801 Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremhmeoimaf1o 17802* The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)

Theoremhmeores 17803 The restriction of a homeomorphism is a homeomorphism. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
t t

Theoremhmeoco 17804 The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Theoremidhmeo 17805 The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
TopOn

Theoremhmeocnvb 17806 The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremhmeoqtop 17807 A homeomorphism is a quotient map. (Contributed by Mario Carneiro, 25-Aug-2015.)
qTop

Theoremhmph 17808 Express the predicate is homeomorph to . (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremhmphi 17809 If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremhmphtop 17810 Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)

Theoremhmphtop1 17811 The relation "being homeomorph to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremhmphtop2 17812 The relation "being homeomorph to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremhmphref 17813 "Is homeomorph to" is reflexive. (Contributed by FL, 25-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Theoremhmphsym 17814 "Is homeomorph to" is symmetric. (Contributed by FL, 8-Mar-2007.) (Proof shortened by Mario Carneiro, 30-May-2014.)

Theoremhmphtr 17815 "Is homeomorph to" is transitive. (Contributed by FL, 9-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremhmpher 17816 "Is homeomorph to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremhmphen 17817 Homeomorphisms preserve the cardinality of the topologies. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremhmphsymb 17818 "Is homeomorph to" is symmetric. (Contributed by FL, 22-Feb-2007.)

Theoremhaushmphlem 17819* Lemma for haushmph 17824 and similar theorems. If the topological property is preserved under injective preimages, then property is preserved under homeomorphisms. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcmphmph 17820 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.)

Theoremconhmph 17821 Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.)

Theoremt0hmph 17822 T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremt1hmph 17823 T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremhaushmph 17824 Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremreghmph 17825 Regularity is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremnrmhmph 17826 Normality is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremhmph0 17827 A topology homeomorphic to the empty set is empty. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremhmphdis 17828 Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.)

Theoremhmphindis 17829 Homeomorphisms preserve topological indiscretion. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremindishmph 17830 Equinumerous sets equipped with their indiscrete topologies are homeomorph (which means in that particular case that a segment is homeomorph to a circle contrary to what Wikipedia claims). (Contributed by FL, 17-Aug-2008.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)

Theoremhmphen2 17831 Homeomorphisms preserve the cardinality of the underlying sets. (Contributed by FL, 17-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremcmphaushmeo 17832 A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. (Contributed by Mario Carneiro, 17-Feb-2015.)

Theoremordthmeolem 17833 Lemma for ordthmeo 17834. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop ordTop

Theoremordthmeo 17834 An order isomorphism is a homeomorphism on the respective order topologies. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop ordTop

Theoremtxhmeo 17835* Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtxswaphmeolem 17836* Show inverse for the "swap components" operation on a cross product. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremtxswaphmeo 17837* There is a homeomorphism from to . (Contributed by Mario Carneiro, 21-Mar-2015.)
TopOn TopOn

Theorempt1hmeo 17838* The canonical homeomorphism from a topological product on a singleton to the topology of the factor. (Contributed by Mario Carneiro, 3-Feb-2015.)
TopOn

Theoremptuncnv 17839* Exhibit the converse function of the map which joins two product topologies on disjoint index sets. (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Theoremptunhmeo 17840* 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 . (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Theoremxpstopnlem1 17841* The function used in xpsval 13797 is a homeomorphism from the binary product topology to the indexed product topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
TopOn       TopOn

Theoremxpstps 17842 A binary product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremxpstopnlem2 17843* Lemma for xpstopn 17844. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremxpstopn 17844 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 17845 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 17846* The inverse of the "currying" function is the uncurrying function. (Contributed by Mario Carneiro, 13-Apr-2015.)
TopOn       TopOn              𝑛Locally        𝑛Locally

Theoremxkohmeo 17847* The Exponential Law for topological spaces. The "currying" function is a homeomorphism on function spaces when and are exponentiable spaces (by xkococn 17692, 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 17848 If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn              qTop

Theoremqtophmeo 17849* 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 17850* 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 17851 A topological space is T0 iff it is homeomorphic to its Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremist1-5lem 17852 Lemma for ist1-5 17854 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 17853 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 17854 A topological space is T1 iff it is both T0 and R0. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremishaus3 17855 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 17856 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 17781. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremreghaus 17857 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 17858 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 17859* Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremelmptrab2 17860* Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremisfbas 17861* 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 17862 There are no empty filter bases. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theorem0nelfb 17863 No filter base contains the empty set. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremfbsspw 17864 A filter base on a set is a subset of the power set. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremfbelss 17865 An element of the filter base is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremfbdmn0 17866 The domain of a filter base is nonempty. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremisfbas2 17867* The predicate " is a filter base." (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfbasssin 17868* A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.)

Theoremfbssfi 17869* 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 17870* 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 17871 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 17872* 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 17873 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 17874* 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 17875 Conditions for the trace of a filter base to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
t t

Theoremtrfbas 17876* 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 17877 Extend class notation with the set of filters on a set.

Definitiondf-fil 17878* 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 17879* The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremfilfbas 17880 A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theorem0nelfil 17881 The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremfileln0 17882 An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.)

Theoremfilsspw 17883 A filter is a subset of the power set of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremfilelss 17884 An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Theoremfilss 17885 A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfilin 17886 A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfiltop 17887 The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremisfil2 17888* Derive the standard axioms of a filter. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Theoremisfildlem 17889* Lemma for isfild 17890. (Contributed by Mario Carneiro, 1-Dec-2013.)

Theoremisfild 17890* 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 17891 A filter is closed under taking intersections. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Theoremfilinn0 17892 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 17893 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 17894 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 17895 The intersection of two filters is a filter. Use fiint 7383 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 17896 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 17897 A filter base on any set is also a filter base on any larger set. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremsnfbas 17898 Condition for a singleton to be a filter base. (Contributed by Stefan O'Rear, 2-Aug-2015.)

Theoremfsubbas 17899 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 17900 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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