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Theorem List for Metamath Proof Explorer - 24801-24900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremerdszelem3 24801* Lemma for erdsze 24810. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem4 24802* Lemma for erdsze 24810. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem5 24803* Lemma for erdsze 24810. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem6 24804* Lemma for erdsze 24810. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem7 24805* Lemma for erdsze 24810. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem8 24806* Lemma for erdsze 24810. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem9 24807* Lemma for erdsze 24810. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem10 24808* Lemma for erdsze 24810. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem11 24809* Lemma for erdsze 24810. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze 24810* The Erdős-Szekeres theorem. For any injective sequence on the reals of length at least , there is either a subsequence of length at least on which is increasing (i.e. a order isomorphism) or a subsequence of length at least on which is decreasing (i.e. a order isomorphism, recalling that is the greater-than relation). (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze2lem1 24811* Lemma for erdsze2 24813. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze2lem2 24812* Lemma for erdsze2 24813. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze2 24813* Generalize the statement of the Erdős-Szekeres theorem erdsze 24810 to "sequences" indexed by an arbitrary subset of , which can be infinite. (Contributed by Mario Carneiro, 22-Jan-2015.)

19.4.6  The Kuratowski closure-complement theorem

Theoremkur14lem1 24814 Lemma for kur14 24824. (Contributed by Mario Carneiro, 17-Feb-2015.)

Theoremkur14lem2 24815 Lemma for kur14 24824. Write interior in terms of closure and complement: where is complement and is closure. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem3 24816 Lemma for kur14 24824. A closure is a subset of the base set. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem4 24817 Lemma for kur14 24824. Complementation is an involution on the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem5 24818 Lemma for kur14 24824. Closure is an idempotent operation in the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem6 24819 Lemma for kur14 24824. If is the complementation operator and is the closure operator, this expresses the identity for any subset of the topological space. This is the key result that lets us cut down long enough sequences of that arise when applying closure and complement repeatedly to , and explains why we end up with a number as large as , yet no larger. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem7 24820 Lemma for kur14 24824: main proof. The set here contains all the distinct combinations of and that can arise, and we prove here that applying or to any element of yields another elemnt of . In operator shorthand, we have . From the identities and , we can reduce any operator combination containing two adjacent identical operators, which is why the list only contains alternating sequences. The reason the sequences don't keep going after a certain point is due to the identity , proved in kur14lem6 24819. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem8 24821 Lemma for kur14 24824. Show that the set contains at most elements. (It could be less if some of the operators take the same value for a given set, but Kuratowski showed that this upper bound of is tight in the sense that there exist topological spaces and subsets of these spaces for which all generated sets are distinct, and indeed the real numbers form such a topological space.) (Contributed by Mario Carneiro, 11-Feb-2015.)
;

Theoremkur14lem9 24822* Lemma for kur14 24824. Since the set is closed under closure and complement, it contains the minimal set as a subset, so also has at most elements. (Indeed , and it's not hard to prove this, but we don't need it for this proof.) (Contributed by Mario Carneiro, 11-Feb-2015.)
;

Theoremkur14lem10 24823* Lemma for kur14 24824. Discharge the set . (Contributed by Mario Carneiro, 11-Feb-2015.)
;

Theoremkur14 24824* Kuratowski's closure-complement theorem. There are at most 14 sets which can be obtained by the application of the closure and complement operations to a set in a topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
;

19.4.7  Retracts and sections

Syntaxcretr 24825 Extend class notation with the retract relation.
Retr

Definitiondf-retr 24826* Define the set of retractions on two topological spaces. We say that is a retraction from to . or Retr iff there is an such that are continuous functions called the retraction and section respectively, and their composite is homotopic to the identity map. If a retraction exists, we say is a retract of . (This terminology is borrowed from HoTT and appears to be nonstandard, although it has similaries to the concept of retract in the category of topological spaces and to a deformation retract in general topology.) Two topological spaces that are retracts of each other are called homotopy equivalent. (Contributed by Mario Carneiro, 11-Feb-2015.)
Retr Htpy

Theoremm1expevenALT 24827 Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)

19.4.8  Path-connected and simply connected spaces

Syntaxcpcon 24828 Extend class notation with the class of path-connected topologies.
PCon

Syntaxcscon 24829 Extend class notation with the class of simply connected topologies.
SCon

Definitiondf-pcon 24830* Define the class of path-connected topologies. A topology is path-connected if there is a path (a continuous function from the unit interval) that goes from to for any points in the space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Definitiondf-scon 24831* Define the class of simply connected topologies. A topology is simply connected if it is path-connected and every loop (continuous path with identical start and endpoint) is contractible to a point (path-homotopic to a constant function). (Contributed by Mario Carneiro, 11-Feb-2015.) (New usage is discouraged.)
SCon PCon

Theoremispcon 24832* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theorempconcn 24833* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theorempcontop 24834 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theoremisscon 24835* The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon PCon

Theoremsconpcon 24836 A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon PCon

Theoremscontop 24837 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon

Theoremsconpht 24838 A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon

Theoremcnpcon 24839 An image of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
PCon PCon

Theorempconcon 24840 A path-connected space is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theoremtxpcon 24841 The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
PCon PCon PCon

Theoremptpcon 24842 The topological product of a collection of path-connected spaces is path-connected. The proof uses the axiom of choice. (Contributed by Mario Carneiro, 17-Feb-2015.)
PCon PCon

Theoremindispcon 24843 The indiscrete topology (or trivial topology) on any set is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 14-Aug-2015.)
PCon

Theoremconpcon 24844 A connected and locally path-connected space is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.)
𝑛Locally PCon PCon

Theoremqtoppcon 24845 A quotient of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
PCon qTop PCon

Theorempconpi1 24846 All fundamental groups in a path-connected space are isomorphic. (Contributed by Mario Carneiro, 12-Feb-2015.)
PCon 𝑔

Theoremsconpht2 24847 Any two paths in a simply connected space with the same start and end point are path-homotopic. (Contributed by Mario Carneiro, 12-Feb-2015.)
SCon

Theoremsconpi1 24848 A path-connected topological space is simply connected iff its fundamental group is trivial. (Contributed by Mario Carneiro, 12-Feb-2015.)
PCon SCon

Theoremtxsconlem 24849 Lemma for txscon 24850. (Contributed by Mario Carneiro, 9-Mar-2015.)

Theoremtxscon 24850 The topological product of two simply connected spaces is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
SCon SCon SCon

Theoremcvxpcon 24851* A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
fld       t        PCon

Theoremcvxscon 24852* A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
fld       t        SCon

Theoremblscon 24853 An open ball in the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
fld              t        SCon

Theoremcnllyscon 24854 The topology of the complex numbers is locally simply connected. (Contributed by Mario Carneiro, 2-Mar-2015.)
fld       Locally SCon

Theoremrescon 24855 A subset of is simply connected iff it is connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
t        SCon

Theoremiooscon 24856 An open interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
t SCon

Theoremiccscon 24857 A closed interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
t SCon

Theoremretopscon 24858 The real numbers are simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
SCon

Theoremiccllyscon 24859 A closed interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
t Locally SCon

Theoremrellyscon 24860 The real numbers are locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
Locally SCon

Theoremiiscon 24861 The unit interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
SCon

Theoremiillyscon 24862 The unit interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
Locally SCon

Theoremiinllycon 24863 The unit interval is locally connected. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝑛Locally

19.4.9  Covering maps

Syntaxccvm 24864 Extend class notation with the class of covering maps.
CovMap

Definitiondf-cvm 24865* Define the class of covering maps on two topological spaces. A function is a covering map if it is continuous and for every point in the target space there is a neighborhood of and a decomposition of the preimage of as a disjoint union such that is a homeomorphism of each set onto . (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap t t

Theoremfncvm 24866 Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmscbv 24867* Change bound variables in the set of even coverings. (Contributed by Mario Carneiro, 17-Feb-2015.)
t t        t t

Theoremiscvm 24868* The property of being a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t               CovMap

Theoremcvmtop1 24869 Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.)
CovMap

Theoremcvmtop2 24870 Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmcn 24871 A covering map is a continuous function. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmcov 24872* Property of a covering map. In order to make the covering property more manageable, we define here the set of all even coverings of an open set in the range. Then the covering property states that every point has a neighborhood which has an even covering. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t               CovMap

Theoremcvmsrcl 24873* Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsi 24874* One direction of cvmsval 24875. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t        t t

Theoremcvmsval 24875* Elementhood in the set of all even coverings of an open set in . is an even covering of if it is a nonempty collection of disjoint open sets in whose union is the preimage of , such that each set is homeomorphic under to . (Contributed by Mario Carneiro, 13-Feb-2015.)
t t        t t

Theoremcvmsss 24876* An even covering is a subset of the topology of the domain (i.e. a collection of open sets). (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsn0 24877* An even covering is nonempty. (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsuni 24878* An even covering of has union equal to the preimage of by . (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsdisj 24879* An even covering of is a disjoint union. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t

Theoremcvmshmeo 24880* Every element of an even covering of is homeomorphic to via . (Contributed by Mario Carneiro, 13-Feb-2015.)
t t        t t

Theoremcvmsf1o 24881* , localized to an element of an even covering of , is a bijection. (Contributed by Mario Carneiro, 14-Feb-2015.)
t t        CovMap

Theoremcvmscld 24882* The sets of an even covering are clopen in the subspace topology on . (Contributed by Mario Carneiro, 14-Feb-2015.)
t t        CovMap t

Theoremcvmsss2 24883* An open subset of an evenly covered set is evenly covered. (Contributed by Mario Carneiro, 7-Jul-2015.)
t t        CovMap

Theoremcvmcov2 24884* The covering map property can be restricted to an open subset. (Contributed by Mario Carneiro, 7-Jul-2015.)
t t        CovMap

Theoremcvmseu 24885* Every element in is a member of a unique element of . (Contributed by Mario Carneiro, 14-Feb-2015.)
t t               CovMap

Theoremcvmsiota 24886* Identify the unique element of containing . (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmopnlem 24887* Lemma for cvmopn 24889. (Contributed by Mario Carneiro, 7-May-2015.)
t t               CovMap

Theoremcvmfolem 24888* Lemma for cvmfo 24909. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t                      CovMap

Theoremcvmopn 24889 A covering map is an open map. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmliftmolem1 24890* Lemma for cvmliftmo 24893. (Contributed by Mario Carneiro, 10-Mar-2015.)
CovMap               𝑛Locally                                           t t                             t

Theoremcvmliftmolem2 24891* Lemma for cvmliftmo 24893. (Contributed by Mario Carneiro, 10-Mar-2015.)
CovMap               𝑛Locally                                           t t

Theoremcvmliftmoi 24892 A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.)
CovMap               𝑛Locally

Theoremcvmliftmo 24893* A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by NM, 17-Jun-2017.)
CovMap               𝑛Locally

Theoremcvmliftlem1 24894* Lemma for cvmlift 24908. In cvmliftlem15 24907, we picked an large enough so that the sections are all contained in an even covering, and the function enumerates these even coverings. So is a neighborhood of , and is an even covering of , which is to say a disjoint union of open sets in whose image is . (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem2 24895* Lemma for cvmlift 24908. is a subset of for each . (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem3 24896* Lemma for cvmlift 24908. Since is a neighborhood of , every element satisfies . (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem4 24897* Lemma for cvmlift 24908. The function will be our lifted path, defined piecewise on each section for . For , it is a "seed" value which makes the rest of the recursion work, a singleton function mapping to . (Contributed by Mario Carneiro, 15-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem5 24898* Lemma for cvmlift 24908. Definition of at a successor. This is a function defined on as where is the unique covering set of that contains evaluated at the last defined point, namely (note that for this is using the seed value ). (Contributed by Mario Carneiro, 15-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem6 24899* Lemma for cvmlift 24908. Induction step for cvmliftlem7 24900. Assuming that is defined at and is a preimage of , the next segment is also defined and is a function on which is a lift for this segment. This follows explicitly from the definition since is in for the entire interval so that maps this into and maps back to . (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem7 24900* Lemma for cvmlift 24908. Prove by induction that every function is well-defined (we can immediately follow this theorem with cvmliftlem6 24899 to show functionality and lifting of ). (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

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