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

Theoremnegcncf 18901* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremnegfcncf 18902* The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)

TheoremabscncfALT 18903 Absolute value is continuous. Alternate proof of abscncf 18884. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) (Proof modification is discouraged.)

Theoremcncfcnvcn 18904 Rewrite cmphaushmeo 17785 for functions on the complexes. (Contributed by Mario Carneiro, 19-Feb-2015.)
fld       t

Theoremcnmptre 18905* Lemma for iirevcn 18908 and related functions. (Contributed by Mario Carneiro, 6-Jun-2014.)
fld       t        t

Theoremcnmpt2pc 18906* Piecewise definition of a continuous function on a real interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
t        t        t                             TopOn

Theoremiirev 18907 Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiirevcn 18908 The reversion function is a continuous map of the unit interval. (Contributed by Mario Carneiro, 6-Jun-2014.)

Theoremiihalf1 18909 Map the first half of into . (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiihalf1cn 18910 The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.)
t

Theoremiihalf2 18911 Map the second half of into . (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiihalf2cn 18912 The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.)
t

Theoremelii1 18913 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)

Theoremelii2 18914 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)

Theoremiimulcl 18915 The unit interval is closed under multiplication. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiimulcn 18916* Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.)

Theoremicoopnst 18917 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 18918 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 18919* The natural bijection from to an arbitrary nontrivial closed interval is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
fld              t

Theoremicopnfcnv 18920* Define a bijection from to . (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremicopnfhmeo 18921* The defined bijection from to is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
fld       t t

Theoremiccpnfcnv 18922* Define a bijection from to . (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremiccpnfhmeo 18923 The defined bijection from to is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
ordTop t

Theoremxrhmeo 18924* 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 18925 The extended reals are homeomorphic to the interval . (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremxrcmp 18926 The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 18790), this means that is a compactification of . (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremxrcon 18927 The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremicccvx 18928 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 18929* 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 18930* 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 18931* The canonical bijection from to described in cnref1o 10563 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 18932* Lemma for cnheibor 18933. (Contributed by Mario Carneiro, 14-Sep-2014.)
fld       t

Theoremcnheibor 18933* 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 18934 The topology on the complexes is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
fld       𝑛Locally

Theoremrellycmp 18935 The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally

Theorembndth 18936* 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 18937* 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 18938* 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 18939* Lemma for lebnum 18942. 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 18940* Lemma for lebnum 18942. As a finite sum of point-to-set distance functions, which are continuous by metdscn 18839, the function is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremlebnumlem3 18941* Lemma for lebnum 18942. 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 18942* 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 18943* Generalize lebnum 18942 to extended metrics. (Contributed by Mario Carneiro, 5-Sep-2015.)

Theoremlebnumii 18944* Specialize the Lebesgue number lemma lebnum 18942 to the unit interval. (Contributed by Mario Carneiro, 14-Feb-2015.)

11.4.12  Path homotopy

Syntaxchtpy 18945 Extend class notation with the class of homotopies between two continuous functions.
Htpy

Syntaxcphtpy 18946 Extend class notation with the class of path homotopies between two continuous functions.

Syntaxcphtpc 18947 Extend class notation with the path homotopy relation.

Definitiondf-htpy 18948* 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 18949* Define the class of path homotopies between two paths ; these are homotopies (in the sense of df-htpy 18948) 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 18950* 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 18951 A homotopy is a continuous function. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                     Htpy

Theoremhtpyi 18952 A homotopy evaluated at its endpoints. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                     Htpy

Theoremishtpyd 18953* Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-2015.)
TopOn                                          Htpy

Theoremhtpycom 18954* Given a homotopy from to , produce a homotopy from to . (Contributed by Mario Carneiro, 23-Feb-2015.)
TopOn                            Htpy        Htpy

Theoremhtpyid 18955* A homotopy from a function to itself. (Contributed by Mario Carneiro, 23-Feb-2015.)
TopOn              Htpy

Theoremhtpyco1 18956* Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
TopOn                            Htpy        Htpy

Theoremhtpyco2 18957 Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
Htpy        Htpy

Theoremhtpycc 18958* Concatenate two homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
TopOn                            Htpy        Htpy        Htpy

Theoremisphtpy 18959* 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 18960 A path homotopy is a homotopy. (Contributed by Mario Carneiro, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
Htpy

Theoremphtpycn 18961 A path homotopy is a continuous function. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpyi 18962 Membership in the class of path homotopies between two continuous functions. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpy01 18963 Two path-homotopic paths have the same start and end point. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremisphtpyd 18964* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
Htpy

Theoremisphtpy2d 18965* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremphtpycom 18966* 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 18967* A homotopy from a path to itself. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)

Theoremphtpyco2 18968 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theoremphtpycc 18969* Concatenate two path homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)

Definitiondf-phtpc 18970* 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 18971 The path homotopy relation is a relation. (Contributed by Mario Carneiro, 7-Jun-2014.) (Revised by Mario Carneiro, 7-Aug-2014.)

Theoremisphtpc 18972 The relation "is path homotopic to". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremphtpcer 18973 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 18974 Path homotopic paths have the same endpoints. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremreparphti 18975* Lemma for reparpht 18976. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)

Theoremreparpht 18976 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 18977 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 6-Jul-2015.)

11.4.13  The fundamental group

Syntaxcpco 18978 Extend class notation with the concatenation operation for paths in a topological space.

Syntaxcomi 18979 Extend class notation with the loop space.

Syntaxcomn 18980 Extend class notation with the higher loop spaces.

Syntaxcpi1 18981 Extend class notation with the fundamental group.

Syntaxcpin 18982 Extend class notation with the higher homotopy groups.

Definitiondf-pco 18983* 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 18984* 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 18985* 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 18986* 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 18987* 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 18988* 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 18989* The concatenation of two paths. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theorempcovalg 18990 Evaluate the concatenation of two paths. (Contributed by Mario Carneiro, 7-Jun-2014.)

Theorempcoval1 18991 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 18992 The starting point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theorempco1 18993 The ending point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theorempcoval2 18994 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 18995 The concatenation of two paths is a path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)

Theoremcopco 18996 The composition of a concatenation of paths with a continuous function. (Contributed by Mario Carneiro, 9-Jul-2015.)

Theorempcohtpylem 18997* Lemma for pcohtpy 18998. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theorempcohtpy 18998 Homotopy invariance of path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theorempcoptcl 18999 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 19000 Concatenation with a point does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

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