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Theorem List for Metamath Proof Explorer - 22901-23000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremerdszelem9 22901* Lemma for erdsze 22904. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem10 22902* Lemma for erdsze 22904. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem11 22903* Lemma for erdsze 22904. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze 22904* 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 22905* Lemma for erdsze2 22907. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze2lem2 22906* Lemma for erdsze2 22907. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze2 22907* Generalize the statement of the Erdős-Szekeres theorem erdsze 22904 to "sequences" indexed by an arbitrary subset of , which can be infinite. (Contributed by Mario Carneiro, 22-Jan-2015.)

16.3.6  The Kuratowski closure-complement theorem

Theoremkur14lem1 22908 Lemma for kur14 22918. (Contributed by Mario Carneiro, 17-Feb-2015.)

Theoremkur14lem2 22909 Lemma for kur14 22918. Write interior in terms of closure and complement: where is complement and is closure. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem3 22910 Lemma for kur14 22918. A closure is a subset of the base set. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem4 22911 Lemma for kur14 22918. Complementation is an involution on the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem5 22912 Lemma for kur14 22918. Closure is an idempotent operation in the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem6 22913 Lemma for kur14 22918. 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 22914 Lemma for kur14 22918: 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 22913. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem8 22915 Lemma for kur14 22918. 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 22916* Lemma for kur14 22918. 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 22917* Lemma for kur14 22918. Discharge the set . (Contributed by Mario Carneiro, 11-Feb-2015.)
;

Theoremkur14 22918* 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.)
;

16.3.7  Retracts and sections

Syntaxcretr 22919 Extend class notation with the retract relation.
Retr

Definitiondf-retr 22920* 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

16.3.8  Path-connected and simply connected spaces

Syntaxcpcon 22921 Extend class notation with the class of path-connected topologies.
PCon

Syntaxcscon 22922 Extend class notation with the class of simply connected topologies.
SCon

Definitiondf-pcon 22923* 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 22924* 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 22925* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theorempconcn 22926* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theorempcontop 22927 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theoremisscon 22928* The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon PCon

Theoremsconpcon 22929 A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon PCon

Theoremscontop 22930 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon

Theoremsconpht 22931 A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon

Theoremcnpcon 22932 An image of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
PCon PCon

Theorempconcon 22933 A path-connected space is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theoremtxpcon 22934 The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
PCon PCon PCon

Theoremptpcon 22935 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 22936 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 22937 A connected and locally path-connected space is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.)
𝑛Locally PCon PCon

Theoremqtoppcon 22938 A quotient of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
PCon qTop PCon

Theorempconpi1 22939 All fundamental groups in a path-connected space are isomorphic. (Contributed by Mario Carneiro, 12-Feb-2015.)
PCon 𝑔

Theoremsconpht2 22940 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 22941 A path-connected topological space is simply connected iff its fundamental group is trivial. (Contributed by Mario Carneiro, 12-Feb-2015.)
PCon SCon

Theoremtxsconlem 22942 Lemma for txscon 22943. (Contributed by Mario Carneiro, 9-Mar-2015.)

Theoremtxscon 22943 The topological product of two simply connected spaces is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
SCon SCon SCon

Theoremcvxpcon 22944* A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
fld       t        PCon

Theoremcvxscon 22945* A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
fld       t        SCon

Theoremblscon 22946 An open ball in the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
fld              t        SCon

Theoremcnllyscon 22947 The topology of the complex numbers is locally simply connected. (Contributed by Mario Carneiro, 2-Mar-2015.)
fld       Locally SCon

Theoremrescon 22948 A subset of is simply connected iff it is connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
t        SCon

Theoremiooscon 22949 An open interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
t SCon

Theoremiccscon 22950 An closed interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
t SCon

Theoremretopscon 22951 The real numbers are simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
SCon

Theoremiccllyscon 22952 An closed interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
t Locally SCon

Theoremrellyscon 22953 The real numbers are locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
Locally SCon

Theoremiiscon 22954 The unit interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
SCon

Theoremiillyscon 22955 The unit interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
Locally SCon

Theoremiinllycon 22956 The unit interval is locally connected. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝑛Locally

16.3.9  Covering maps

Syntaxccvm 22957 Extend class notation with the class of covering maps.
CovMap

Definitiondf-cvm 22958* 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 22959 Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmscbv 22960* Change bound variables in the set of even coverings. (Contributed by Mario Carneiro, 17-Feb-2015.)
t t        t t

Theoremiscvm 22961* The property of being a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t               CovMap

Theoremcvmtop1 22962 Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.)
CovMap

Theoremcvmtop2 22963 Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmcn 22964 A covering map is a continuous function. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmcov 22965* 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 22966* Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsi 22967* One direction of cvmsval 22968. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t        t t

Theoremcvmsval 22968* 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 22969* 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 22970* An even covering is nonempty. (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsuni 22971* An even covering of has union equal to the preimage of by . (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsdisj 22972* An even covering of is a disjoint union. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t

Theoremcvmshmeo 22973* Every element of an even covering of is homeomorphic to via . (Contributed by Mario Carneiro, 13-Feb-2015.)
t t        t t

Theoremcvmsf1o 22974* , localized to an element of an even covering of , is a bijection. (Contributed by Mario Carneiro, 14-Feb-2015.)
t t        CovMap

Theoremcvmscld 22975* 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 22976* An open subset of an evenly covered set is evenly covered. (Contributed by Mario Carneiro, 7-Jul-2015.)
t t        CovMap

Theoremcvmcov2 22977* The covering map property can be restricted to an open subset. (Contributed by Mario Carneiro, 7-Jul-2015.)
t t        CovMap

Theoremcvmseu 22978* Every element in is a member of a unique element of . (Contributed by Mario Carneiro, 14-Feb-2015.)
t t               CovMap

Theoremcvmsiota 22979* Identify the unique element of containing . (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmopnlem 22980* Lemma for cvmopn 22982. (Contributed by Mario Carneiro, 7-May-2015.)
t t               CovMap

Theoremcvmfolem 22981* Lemma for cvmfo 23002. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t                      CovMap

Theoremcvmopn 22982 A covering map is an open map. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmliftmolem1 22983* Lemma for cvmliftmo 22986. (Contributed by Mario Carneiro, 10-Mar-2015.)
CovMap               𝑛Locally                                           t t                             t

Theoremcvmliftmolem2 22984* Lemma for cvmliftmo 22986. (Contributed by Mario Carneiro, 10-Mar-2015.)
CovMap               𝑛Locally                                           t t

Theoremcvmliftmoi 22985 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 22986* 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

Theoremcvmliftlem1 22987* Lemma for cvmlift 23001. In cvmliftlem15 23000, 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 22988* Lemma for cvmlift 23001. is a subset of for each . (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem3 22989* Lemma for cvmlift 23001. Since is a neighborhood of , every element satisfies . (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem4 22990* Lemma for cvmlift 23001. 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 22991* Lemma for cvmlift 23001. 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 22992* Lemma for cvmlift 23001. Induction step for cvmliftlem7 22993. 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 22993* Lemma for cvmlift 23001. Prove by induction that every function is well-defined (we can immediately follow this theorem with cvmliftlem6 22992 to show functionality and lifting of ). (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem8 22994* Lemma for cvmlift 23001. The functions are continuous functions because they are defined as where is continuous and is a homeomorphism. (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap                                                                       t

Theoremcvmliftlem9 22995* Lemma for cvmlift 23001. The functions are defined on almost disjoint intervals, but they overlap at the edges. Here we show that at these points the functions agree on their common domain. (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem10 22996* Lemma for cvmlift 23001. The function is going to be our complete lifted path, formed by unioning together all the functions (each of which is defined on one segment of the interval). Here we prove by induction that is a continuous function and a lift of by applying cvmliftlem6 22992, cvmliftlem7 22993 (to show it is a function and a lift), cvmliftlem8 22994 (to show it is continuous), and cvmliftlem9 22995 (to show that different functions agree on the intersection of their domains, so that the pasting lemma paste 16854 gives that is well-defined and continuous). (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap                                                                       t        t

Theoremcvmliftlem11 22997* Lemma for cvmlift 23001. (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem13 22998* Lemma for cvmlift 23001. The initial value of is because is a subset of which takes value at . (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem14 22999* Lemma for cvmlift 23001. Putting the results of cvmliftlem11 22997, cvmliftlem13 22998 and cvmliftmo 22986 together, we have that is a continuous function, satisfies and , and is equal to any other function which also has these properties, so it follows that is the unique lift of . (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem15 23000* Lemma for cvmlift 23001. Discharge the assumptions of cvmliftlem14 22999. The set of all open subsets of the unit interval such that is contained in an even covering of some open set in is a cover of by the definition of a covering map, so by the Lebesgue number lemma lebnumii 18296, there is a subdivision of the unit interval into equal parts such that each part is entirely contained within one such open set of . Then using finite choice ac6sfi 6986 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 22999. (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

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