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Theorem List for Metamath Proof Explorer - 18901-19000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiimulcl 18901 The unit interval is closed under multiplication. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiimulcn 18902* Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.)

Theoremicoopnst 18903 A half-open interval starting at is open in the closed interval from to . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)

Theoremiocopnst 18904 A half-open interval ending at is open in the closed interval from to . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)

Theoremicchmeo 18905* The natural bijection from to an arbitrary nontrivial closed interval is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
fld              t

Theoremicopnfcnv 18906* Define a bijection from to . (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremicopnfhmeo 18907* The defined bijection from to is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
fld       t t

Theoremiccpnfcnv 18908* Define a bijection from to . (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremiccpnfhmeo 18909 The defined bijection from to is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
ordTop t

Theoremxrhmeo 18910* The bijection from to the extended reals is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
fld       ordTop        t ordTop

Theoremxrhmph 18911 The extended reals are homeomorphic to the interval . (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremxrcmp 18912 The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 18776), this means that is a compactification of . (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremxrcon 18913 The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremicccvx 18914 A linear combination of two reals lies in the interval between them. Equivalently, a closed interval is a convex set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremoprpiece1res1 18915* Restriction to the first part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)

Theoremoprpiece1res2 18916* Restriction to the second part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)

Theoremcnrehmeo 18917* The canonical bijection from to described in cnref1o 10553 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.)
fld

Theoremcnheiborlem 18918* Lemma for cnheibor 18919. (Contributed by Mario Carneiro, 14-Sep-2014.)
fld       t

Theoremcnheibor 18919* Heine-Borel theorem for complex numbers. A subset of is compact iff it is closed and bounded. (Contributed by Mario Carneiro, 14-Sep-2014.)
fld       t

Theoremcnllycmp 18920 The topology on the complexes is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
fld       𝑛Locally

Theoremrellycmp 18921 The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally

Theorembndth 18922* The Boundedness Theorem. A continuous function from a compact topological space to the reals is bounded (above). (Boundedness below is obtained by applying this theorem to .) (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremevth 18923* The Extreme Value Theorem. A continuous function from a nonempty compact topological space to the reals attains its maximum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremevth2 18924* The Extreme Value Theorem, minimum version. A continuous function from a nonempty compact topological space to the reals attains its minimum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)

Theoremlebnumlem1 18925* Lemma for lebnum 18928. The function measures the sum of all of the distances to escape the sets of the cover. Since by assumption it is a cover, there is at least one set which covers a given point, and since it is open, the point is a positive distance from the edge of the set. Thus, the sum is a strictly positive number. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremlebnumlem2 18926* Lemma for lebnum 18928. As a finite sum of point-to-set distance functions, which are continuous by metdscn 18825, the function is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremlebnumlem3 18927* Lemma for lebnum 18928. By the previous lemmas, is continuous and positive on a compact set, so it has a positive minimum . Then setting , since for each we have iff , if for all then summing over yields , in contradiction to the assumption that is the minimum of . (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremlebnum 18928* The Lebesgue number lemma, or Lebesgue covering lemma. If is a compact metric space and is an open cover of , then there exists a positive real number such that every ball of size (and every subset of a ball of size , including every subset of diameter less than ) is a subset of some member of the cover. (Contributed by Mario Carneiro, 14-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Sep-2015.)

Theoremxlebnum 18929* Generalize lebnum 18928 to extended metrics. (Contributed by Mario Carneiro, 5-Sep-2015.)

Theoremlebnumii 18930* Specialize the Lebesgue number lemma lebnum 18928 to the unit interval. (Contributed by Mario Carneiro, 14-Feb-2015.)

11.4.12  Path homotopy

Syntaxchtpy 18931 Extend class notation with the class of homotopies between two continuous functions.
Htpy

Syntaxcphtpy 18932 Extend class notation with the class of path homotopies between two continuous functions.

Syntaxcphtpc 18933 Extend class notation with the path homotopy relation.

Definitiondf-htpy 18934* Define the function which takes topological spaces and two continuous functions and returns the class of homotopies from to . (Contributed by Mario Carneiro, 22-Feb-2015.)
Htpy

Definitiondf-phtpy 18935* Define the class of path homotopies between two paths ; these are homotopies (in the sense of df-htpy 18934) which also preserve both endpoints of the paths throughout the homotopy. Definition of [Hatcher] p. 25. (Contributed by Jeff Madsen, 2-Sep-2009.)
Htpy

Theoremishtpy 18936* Membership in the class of homotopies between two continuous functions. (Contributed by Mario Carneiro, 22-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
TopOn                     Htpy

Theoremhtpycn 18937 A homotopy is a continuous function. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                     Htpy

Theoremhtpyi 18938 A homotopy evaluated at its endpoints. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                     Htpy

Theoremishtpyd 18939* Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                                          Htpy

Theoremhtpycom 18940* Given a homotopy from to , produce a homotopy from to . (Contributed by Mario Carneiro, 23-Feb-2015.)
TopOn                            Htpy        Htpy

Theoremhtpyid 18941* A homotopy from a function to itself. (Contributed by Mario Carneiro, 23-Feb-2015.)
TopOn              Htpy

Theoremhtpyco1 18942* Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
TopOn                            Htpy        Htpy

Theoremhtpyco2 18943 Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
Htpy        Htpy

Theoremhtpycc 18944* Concatenate two homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
TopOn                            Htpy        Htpy        Htpy

Theoremisphtpy 18945* Membership in the class of path homotopies between two continuous functions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
Htpy

Theoremphtpyhtpy 18946 A path homotopy is a homotopy. (Contributed by Mario Carneiro, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
Htpy

Theoremphtpycn 18947 A path homotopy is a continuous function. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpyi 18948 Membership in the class of path homotopies between two continuous functions. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpy01 18949 Two path-homotopic paths have the same start and end point. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremisphtpyd 18950* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
Htpy

Theoremisphtpy2d 18951* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpycom 18952* Given a homotopy from to , produce a homotopy from to . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)

Theoremphtpyid 18953* A homotopy from a path to itself. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)

Theoremphtpyco2 18954 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theoremphtpycc 18955* Concatenate two path homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)

Definitiondf-phtpc 18956* Define the function which takes a topology and returns the path homotopy relation on that topology. Definition of [Hatcher] p. 25. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)

Theoremphtpcrel 18957 The path homotopy relation is a relation. (Contributed by Mario Carneiro, 7-Jun-2014.) (Revised by Mario Carneiro, 7-Aug-2014.)

Theoremisphtpc 18958 The relation "is path homotopic to". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremphtpcer 18959 Path homotopy is an equivalence relation. Proposition 1.2 of [Hatcher] p. 26. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 6-Jul-2015.)

Theoremphtpc01 18960 Path homotopic paths have the same endpoints. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremreparphti 18961* Lemma for reparpht 18962. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)

Theoremreparpht 18962 Reparametrization lemma. The reparametrization of a path by any continuous map with and is path-homotopic to the original path. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Feb-2015.)

Theoremphtpcco2 18963 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 6-Jul-2015.)

11.4.13  The fundamental group

Syntaxcpco 18964 Extend class notation with the concatenation operation for paths in a topological space.

Syntaxcomi 18965 Extend class notation with the loop space.

Syntaxcomn 18966 Extend class notation with the higher loop spaces.

Syntaxcpi1 18967 Extend class notation with the fundamental group.

Syntaxcpin 18968 Extend class notation with the higher homotopy groups.

Definitiondf-pco 18969* Define the concatenation of two paths in a topological space . For simplicity of definition, we define it on all paths, not just those whose endpoints line up. Definition of [Hatcher] p. 26. Hatcher denotes path concatenation with a square dot; other authors, such as Munkres, use a star. (Contributed by Jeff Madsen, 15-Jun-2010.)

Definitiondf-om1 18970* Define the loop space of a topological space, with a magma structure on it given by concatenation of loops. This structure is not a group, but the operation is compatible with homotopy, which allows the homotopy groups to be defined based on this operation. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopSet

Definitiondf-omn 18971* Define the n-th iterated loop space of a topological space. Unlike this is actually a pointed topological space, which is to say a tuple of a topological space (a member of , not ) and a point in the space. Higher loop spaces select the constant loop at the point from the lower loop space for the distinguished point. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopSet

Definitiondf-pi1 18972* Define the fundamental group, whose operation is given by concatenation of homotopy classes of loops. Definition of [Hatcher] p. 26. (Contributed by Mario Carneiro, 11-Feb-2015.)
s

Definitiondf-pin 18973* Define the n-th homotopy group, which is formed by taking the -th loop space and forming the quotient under the relation of path homotopy equivalence in the base space of the -th loop space, which is the -th loop space. For , since this is not well-defined we replace this relation with the path-connectedness relation, so that the -th homotopy group is the set of path components of . (Since the -th loop space does not have a group operation, neither does the -th homotopy group, but the rest are genuine groups.) (Contributed by Mario Carneiro, 11-Feb-2015.)
s

Theorempcofval 18974* The value of the path concatenation function on a topological space. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)

Theorempcoval 18975* The concatenation of two paths. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theorempcovalg 18976 Evaluate the concatenation of two paths. (Contributed by Mario Carneiro, 7-Jun-2014.)

Theorempcoval1 18977 Evaluate the concatenation of two paths on the first half. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)

Theorempco0 18978 The starting point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theorempco1 18979 The ending point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theorempcoval2 18980 Evaluate the concatenation of two paths on the second half. (Contributed by Jeff Madsen, 15-Jun-2010.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)

Theorempcocn 18981 The concatenation of two paths is a path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)

Theoremcopco 18982 The composition of a concatenation of paths with a continuous function. (Contributed by Mario Carneiro, 9-Jul-2015.)

Theorempcohtpylem 18983* Lemma for pcohtpy 18984. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theorempcohtpy 18984 Homotopy invariance of path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theorempcoptcl 18985 A constant function is a path from to itself. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 19-Mar-2015.)
TopOn

Theorempcopt 18986 Concatenation with a point does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theorempcopt2 18987 Concatenation with a point does not affect homotopy class. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcoass 18988* Order of concatenation does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)

Theorempcorevcl 18989* Closure for a reversed path. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcorevlem 18990* Lemma for pcorev 18991. Prove continuity of the homotopy function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)

Theorempcorev 18991* Concatenation with the reverse path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theorempcorev2 18992* Concatenation with the reverse path. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theorempcophtb 18993* The path homotopy equivalence relation on two paths with the same start and end point can be written in terms of the loop formed by concatenating with the inverse of . Thus, all the homotopy information in is available if we restrict our attention to closed loops, as in the definition of the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.)

Theoremom1val 18994* The definition of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn              TopSet

Theoremom1bas 18995* The base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremom1elbas 18996 Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremom1addcl 18997 Closure of the group operation of the loop space. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 5-Sep-2015.)
TopOn

Theoremom1plusg 18998 The group operation (which isn't much more than a magma) of the loop space. (Contributed by Mario Carneiro, 11-Feb-2015.)
TopOn

Theoremom1tset 18999 The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn              TopSet

Theoremom1opn 19000 The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn                            t

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