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Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremkur14lem8 24901 Lemma for kur14 24904. 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.)
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Theoremkur14lem9 24902* Lemma for kur14 24904. 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 24903* Lemma for kur14 24904. Discharge the set . (Contributed by Mario Carneiro, 11-Feb-2015.)
;

Theoremkur14 24904* 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 24905 Extend class notation with the retract relation.
Retr

Definitiondf-retr 24906* 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 24907 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 24908 Extend class notation with the class of path-connected topologies.
PCon

Syntaxcscon 24909 Extend class notation with the class of simply connected topologies.
SCon

Definitiondf-pcon 24910* 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 24911* 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 24912* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theorempconcn 24913* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theorempcontop 24914 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theoremisscon 24915* The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon PCon

Theoremsconpcon 24916 A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon PCon

Theoremscontop 24917 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon

Theoremsconpht 24918 A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon

Theoremcnpcon 24919 An image of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
PCon PCon

Theorempconcon 24920 A path-connected space is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theoremtxpcon 24921 The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
PCon PCon PCon

Theoremptpcon 24922 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 24923 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 24924 A connected and locally path-connected space is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.)
𝑛Locally PCon PCon

Theoremqtoppcon 24925 A quotient of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
PCon qTop PCon

Theorempconpi1 24926 All fundamental groups in a path-connected space are isomorphic. (Contributed by Mario Carneiro, 12-Feb-2015.)
PCon 𝑔

Theoremsconpht2 24927 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 24928 A path-connected topological space is simply connected iff its fundamental group is trivial. (Contributed by Mario Carneiro, 12-Feb-2015.)
PCon SCon

Theoremtxsconlem 24929 Lemma for txscon 24930. (Contributed by Mario Carneiro, 9-Mar-2015.)

Theoremtxscon 24930 The topological product of two simply connected spaces is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
SCon SCon SCon

Theoremcvxpcon 24931* A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
fld       t        PCon

Theoremcvxscon 24932* A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
fld       t        SCon

Theoremblscon 24933 An open ball in the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
fld              t        SCon

Theoremcnllyscon 24934 The topology of the complex numbers is locally simply connected. (Contributed by Mario Carneiro, 2-Mar-2015.)
fld       Locally SCon

Theoremrescon 24935 A subset of is simply connected iff it is connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
t        SCon

Theoremiooscon 24936 An open interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
t SCon

Theoremiccscon 24937 A closed interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
t SCon

Theoremretopscon 24938 The real numbers are simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
SCon

Theoremiccllyscon 24939 A closed interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
t Locally SCon

Theoremrellyscon 24940 The real numbers are locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
Locally SCon

Theoremiiscon 24941 The unit interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
SCon

Theoremiillyscon 24942 The unit interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
Locally SCon

Theoremiinllycon 24943 The unit interval is locally connected. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝑛Locally

19.4.9  Covering maps

Syntaxccvm 24944 Extend class notation with the class of covering maps.
CovMap

Definitiondf-cvm 24945* 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 24946 Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmscbv 24947* Change bound variables in the set of even coverings. (Contributed by Mario Carneiro, 17-Feb-2015.)
t t        t t

Theoremiscvm 24948* The property of being a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t               CovMap

Theoremcvmtop1 24949 Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.)
CovMap

Theoremcvmtop2 24950 Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmcn 24951 A covering map is a continuous function. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmcov 24952* 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 24953* Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsi 24954* One direction of cvmsval 24955. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t        t t

Theoremcvmsval 24955* 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 24956* 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 24957* An even covering is nonempty. (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsuni 24958* An even covering of has union equal to the preimage of by . (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsdisj 24959* An even covering of is a disjoint union. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t

Theoremcvmshmeo 24960* Every element of an even covering of is homeomorphic to via . (Contributed by Mario Carneiro, 13-Feb-2015.)
t t        t t

Theoremcvmsf1o 24961* , localized to an element of an even covering of , is a bijection. (Contributed by Mario Carneiro, 14-Feb-2015.)
t t        CovMap

Theoremcvmscld 24962* 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 24963* An open subset of an evenly covered set is evenly covered. (Contributed by Mario Carneiro, 7-Jul-2015.)
t t        CovMap

Theoremcvmcov2 24964* The covering map property can be restricted to an open subset. (Contributed by Mario Carneiro, 7-Jul-2015.)
t t        CovMap

Theoremcvmseu 24965* Every element in is a member of a unique element of . (Contributed by Mario Carneiro, 14-Feb-2015.)
t t               CovMap

Theoremcvmsiota 24966* Identify the unique element of containing . (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmopnlem 24967* Lemma for cvmopn 24969. (Contributed by Mario Carneiro, 7-May-2015.)
t t               CovMap

Theoremcvmfolem 24968* Lemma for cvmfo 24989. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t                      CovMap

Theoremcvmopn 24969 A covering map is an open map. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmliftmolem1 24970* Lemma for cvmliftmo 24973. (Contributed by Mario Carneiro, 10-Mar-2015.)
CovMap               𝑛Locally                                           t t                             t

Theoremcvmliftmolem2 24971* Lemma for cvmliftmo 24973. (Contributed by Mario Carneiro, 10-Mar-2015.)
CovMap               𝑛Locally                                           t t

Theoremcvmliftmoi 24972 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 24973* 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 24974* Lemma for cvmlift 24988. In cvmliftlem15 24987, 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 24975* Lemma for cvmlift 24988. is a subset of for each . (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem3 24976* Lemma for cvmlift 24988. Since is a neighborhood of , every element satisfies . (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem4 24977* Lemma for cvmlift 24988. 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 24978* Lemma for cvmlift 24988. 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 24979* Lemma for cvmlift 24988. Induction step for cvmliftlem7 24980. 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 24980* Lemma for cvmlift 24988. Prove by induction that every function is well-defined (we can immediately follow this theorem with cvmliftlem6 24979 to show functionality and lifting of ). (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem8 24981* Lemma for cvmlift 24988. 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 24982* Lemma for cvmlift 24988. 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 24983* Lemma for cvmlift 24988. 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 24979, cvmliftlem7 24980 (to show it is a function and a lift), cvmliftlem8 24981 (to show it is continuous), and cvmliftlem9 24982 (to show that different functions agree on the intersection of their domains, so that the pasting lemma paste 17360 gives that is well-defined and continuous). (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap                                                                       t        t

Theoremcvmliftlem11 24984* Lemma for cvmlift 24988. (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem13 24985* Lemma for cvmlift 24988. 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 24986* Lemma for cvmlift 24988. Putting the results of cvmliftlem11 24984, cvmliftlem13 24985 and cvmliftmo 24973 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 24987* Lemma for cvmlift 24988. Discharge the assumptions of cvmliftlem14 24986. 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 18993, 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 7353 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 24986. (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmlift 24988* One of the important properties of covering maps is that any path in the base space "lifts" to a path in the covering space such that , and given a starting point in the covering space this lift is unique. The proof is contained in cvmliftlem1 24974 thru cvmliftlem15 24987. (Contributed by Mario Carneiro, 16-Feb-2015.)
CovMap

Theoremcvmfo 24989 A covering map is an onto function. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmliftiota 24990* Write out a function that is the unique lift of . (Contributed by Mario Carneiro, 16-Feb-2015.)
CovMap

Theoremcvmlift2lem1 24991* Lemma for cvmlift2 25005. (Contributed by Mario Carneiro, 1-Jun-2015.)

Theoremcvmlift2lem9a 24992* Lemma for cvmlift2 25005 and cvmlift3 25017. (Contributed by Mario Carneiro, 9-Jul-2015.)
t t        CovMap                                                                t

Theoremcvmlift2lem2 24993* Lemma for cvmlift2 25005. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmlift2lem3 24994* Lemma for cvmlift2 25005. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmlift2lem4 24995* Lemma for cvmlift2 25005. (Contributed by Mario Carneiro, 1-Jun-2015.)
CovMap

Theoremcvmlift2lem5 24996* Lemma for cvmlift2 25005. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmlift2lem6 24997* Lemma for cvmlift2 25005. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap                                           t

Theoremcvmlift2lem7 24998* Lemma for cvmlift2 25005. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmlift2lem8 24999* Lemma for cvmlift2 25005. (Contributed by Mario Carneiro, 9-Mar-2015.)
CovMap

Theoremcvmlift2lem9 25000* Lemma for cvmlift2 25005. (Contributed by Mario Carneiro, 1-Jun-2015.)
CovMap                                           t t                                    t        t                                    t               t

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