Home Metamath Proof ExplorerTheorem List (p. 257 of 328) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21514) Hilbert Space Explorer (21515-23037) Users' Mathboxes (23038-32776)

Theorem List for Metamath Proof Explorer - 25601-25700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-mmat 25601* Matrix multiplication. Meaningful if is a ring at least. here should be (to be traditional). But in set.mm is oriented and has a limit definition embedded and thus doesn't fit the needs of this generic definition. Experimental. (Contributed by FL, 29-Aug-2010.)

18.13.25  Affine spaces

Syntaxcraffsp 25602 Extend class notation with the class of all R affine spaces.

Definitiondf-raffsp 25603* Define a affine space id est a vector space ( called the free vectors class ) together with a function . associates to each vector a bijection from a set (called the space) to itself (here is retrieved from the operation.) Technically speaking, is a faithful (i.e. injective) and transitive group action (id est a group homomorphism whose range is the underlying set of a symmetry group ). Informally speaking the aim of all of that is to associate to each point of a unique point of through the "action" of a vector of and thus to formalize the idea of translation. When we have embedded the idea of translation it is easy to define a repere and thus all the cartesian geometry is available. (Contributed by FL, 29-Aug-2010.)
GrpOpHom

18.13.26  Intervals of reals and extended reals

Theorembsi 25604* Membership to the set of open intervals implied the existence of two bounds in the set of the extended reals. (Contributed by FL, 31-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioo1t3 25605 If an open interval has an element, then . (Contributed by FL, 14-Aug-2007.)

Theoremoisbmi 25606 An open interval with its upper bound equal to is empty. (Contributed by FL, 12-Sep-2007.)

Theoremoisbmj 25607 An open interval with its lower bound equal to is empty. (Contributed by FL, 12-Sep-2007.)

Theoremtruni1 25608 Closure of translation in a half-infinite interval. (Contributed by FL, 11-Sep-2007.)

Theoremtruni2 25609 Closure of translation in a half-infinite interval. (Contributed by FL, 26-Jan-2009.)

Theoremtruni3 25610 Closure of translation in a half-infinite interval. (Contributed by FL, 26-Jan-2009.)

Theoremcbci 25611 The center belongs to a centered interval. (Contributed by FL, 5-Jan-2009.)

Theoremoibbi1 25612 An open interval is included in a bound below interval. (Contributed by FL, 26-Jan-2009.) (Revised by Mario Carneiro, 3-May-2015.)

Theoremoibbi2 25613 An open interval is included in a bound above interval. (Contributed by FL, 26-Jan-2009.)

Theoremnelioo5 25614 Membership in an open interval of extended reals. (Contributed by FL, 7-Dec-2010.)

18.13.27  Topology

Theoremtopnem 25615 A topology is not empty. (Contributed by FL, 1-Jun-2008.)

Theoremclsint 25616 The closure of an intersection is included in the intersection of the closures. (Contributed by FL, 23-Feb-2009.)

Theoremislp3 25617* The predicate " is a limit point of " in terms of open sets. see islp2 16893, elcls 16826, islp 16888. (Contributed by FL, 31-Jul-2009.)

Theoreminttop2 25618* The intersection of a family of topologies is a topology. (Contributed by FL, 19-Sep-2011.)

Theoreminttop3 25619 The intersection of a family of topologies is a topology. (Contributed by FL, 19-Sep-2011.)

Theoreminttop4 25620 The intersection of two topologies is a topology. (Contributed by FL, 19-Sep-2011.)

Theoremunint2t 25621 The intersection of two topologies over the same underlying set is a topology over . compare uniin 3863. (Contributed by FL, 27-Nov-2011.)

Theoremintfmu2 25622* The intersection of a family of topologies over the same underying set is a topology over . (Contributed by FL, 27-Nov-2011.)

Theoremapnei 25623* Any point has a neighborhood. (Contributed by FL, 15-Oct-2012.)

Theoremnpmp 25624 A neighborhood of a point can't be empty. (Contributed by FL, 15-Oct-2012.)

Theorembasexre 25625 A basis for the standard topology over the extended reals. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)

Theoremstovr 25626 The standard topology over . (Contributed by FL, 15-Sep-2013.)

Theoremcldifemp 25627 The closure of a class is empty iff is empty. (Contributed by FL, 15-Sep-2013.)

18.13.28  Continuous functions

Theoremcnrsfin 25628 A mapping remains continuous when the topology associated to its domain is replaced by a finer one. (Contributed by FL, 22-May-2008.)

Theoremcnrscoa 25629 A mapping remains continuous when the topology associated to its range is replaced by a coarser one. (Contributed by FL, 1-Jun-2008.)

Theoremmapdiscn 25630 Any mapping whose domain is associated to the discrete topology is continuous. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Mario Carneiro, 7-Apr-2015.)

Theoremmapudiscn 25631 Any mapping whose range is associated to the undiscrete topology is continuous. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremsallnei 25632* Two ways to state the set of all the neighborhoods. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremnsn 25633* The neighborhoods of the singletons are neighborhoods. (Contributed by FL, 2-Aug-2009.)

Theoremosneisi 25634* The non-empty open sets are neighborhoods of the singletons. (Contributed by FL, 16-Jul-2009.)

Theoremelsubops 25635 The elements of a subbase are open sets. (Contributed by FL, 16-Apr-2012.) (Revised by Mario Carneiro, 14-Dec-2013.)

18.13.29  Homeomorphisms

Theoremdmhmph 25636 is a relation whose domain is included in . (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)

Theoremrnhmph 25637 is a relation whose range is included in . (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)

Theoremhmeogrplem 25638* Lemma for hmeogrp 25640. (Contributed by FL, 30-May-2014.)

Theoremhmeogrpi 25639* Lemma for hmeogrp 25640. (Contributed by FL, 31-May-2014.)

Theoremhmeogrp 25640* Homeomorphisms on a topology is a group for composition. This means from Felix Klein's point of view that a set equipped with a topology is a geometry, namely the so-called rubber sheet geometry. (Contributed by FL, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 31-May-2014.)

18.13.30  Initial and final topologies

Theoremintopcoaconlem3b 25641* The underlying set of the initial topology is the domain of the mappings . (Contributed by FL, 24-Apr-2012.) (Revised by Mario Carneiro, 25-Nov-2013.)

Theoremintopcoaconlem3 25642* The underlying set of the initial topology is the domain of the mappings . (Contributed by FL, 21-Apr-2012.) (Revised by Mario Carneiro, 25-Nov-2013.)

Theoremintopcoaconb 25643* The initial topology is the coarsest one making the functions continuous . (Contributed by FL, 14-May-2012.) (Revised by Mario Carneiro, 26-Nov-2013.)

Theoremintopcoaconc 25644* The initial topology makes the functions continuous. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 26-Nov-2013.)

Theoremqusp 25645* A quotient space is a topology. (Contributed by FL, 4-Jun-2007.)

Theoremintcont 25646 If is continous over two topologies and then it is continuous over . (Contributed by FL, 27-Nov-2011.)

Syntaxctopx 25647 Extend class notation with a function whose value is a product topology.

Definitiondf-prtop 25648* The product topology of a family of topologies is the coarsest topology over the product of the underlying sets that makes the projections continuous. (Bourbaki TG I.14 ex. 3) Experimental. (Contributed by FL, 4-Dec-2011.)

Theoremusptoplem 25649* Lemma for usptop 25653. (Contributed by FL, 5-Dec-2011.) (Revised by Mario Carneiro, 26-Jan-2015.)

Theoremistopx 25650* Definition of the product topology of a family of topologies . (Contributed by FL, 4-Dec-2011.) (Revised by Mario Carneiro, 26-Jan-2015.)

Theoremistopxc 25651* Product of topology . (Contributed by FL, 15-Sep-2013.)

Theoremprtoptop 25652 The product topology of a family of topologies is a topology. (Contributed by FL, 5-Dec-2011.) (Proof shortened by Mario Carneiro, 26-Jan-2015.)

Theoremusptop 25653* The underlying set of a product topology. (Contributed by FL, 5-Dec-2011.)

Theoremprcnt 25654* The projections are continuous. (Contributed by FL, 18-Apr-2012.)

18.13.31  Filters

Theoremefilcp 25655* A filter containing a set exists iff has the finite intersection property (i.e. no finite intersection of elements of is empty). Bourbaki TG I.37 prop. 1. (Contributed by FL, 20-Nov-2007.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Theoremfilint2 25656 A filter is closed under taking finite intersections. (Contributed by FL, 27-Apr-2008.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremfisub 25657* If a set has the finite intersection property, its subsets have also this property. (Contributed by FL, 27-Apr-2008.)

Theoremfgsb2 25658* Filter generated by a subbasis . Bourbaki TG I.37 paragraph above prop. 1. (The theorem has been slightly modified because the definitions of the empty set are different in Bourbaki and Metamath.) (Contributed by FL, 27-Apr-2008.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremcnfilca 25659* Condition to have a filter finer than a given filter and containing a set . Bourbaki T.G. I.37 cor. 1 (Contributed by FL, 27-Apr-2008.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Theoremfil2ss 25660* A condition for a filter to be finer than another filter. Compare fgss2 17585. (Contributed by FL, 8-Aug-2011.) (Revised by Mario Carneiro, 6-Aug-2015.)

18.13.32  Limits

Theoremplimfil 25661 The predicate "is a limit of a filter". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)

Theoremlimvinlv 25662 The limit value of a convergent function whose values are in a Hausdorff space belongs to the set of the limit values. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Theoremflfneih 25663* A neighborhood of the limit value of a convergent function whose values are in a Hausdorff space contains the image of a filter element. (Contributed by FL, 18-Nov-2010.) (Revised by Mario Carneiro, 6-Aug-2015.)

Theoremlimfilnei 25664 is a limit of the filter of the neighborhoods of . (Contributed by FL, 27-May-2011.) (Proof shortened by Mario Carneiro, 9-Apr-2015.)

Theoremconttnf2 25665 is continous at point iff is a limit of the image filter of the neighborhoods of . (Contributed by FL, 7-Aug-2011.) (Revised by Mario Carneiro, 6-Aug-2015.)

Theoremiscnp4 25666* The predicate " is a continuous function from topology to topology at point ." in terms of neighborhoods. (Contributed by FL, 18-Jul-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
TopOn TopOn

Theoremcnpflf4 25667 If is continuous at point , and the filter base converges to then converges to . Bourbaki TG I.50 cor 1. (Contributed by FL, 19-Sep-2011.) (Revised by Stefan O'Rear, 7-Aug-2015.)

Theoremlimfn 25668 The limits of a function are elements of its range. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Aug-2015.)

Theoremlimfn2 25669 If is a limit of a function , is an element of the range of . (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 29-Jul-2015.)

Theoremlimfn3 25670 If is the limit of a convergent function in a Hausdorff space, is an element of the range of the function. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 7-Aug-2015.)

Theoremcmptdst 25671 tends to if is continuous at point and tends to A . Bourbaki TG I.50 cor. 2. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 8-Aug-2015.)

Theoremunexun 25672* If is an element of and has a unique element, . (Contributed by FL, 15-Oct-2012.)

Theoremlimhun 25673 In a Hausdorff space if is a limit of a convergent function , then is the unique limit of . (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 8-Aug-2015.)

Theoremcmptdst2 25674 tends to if is continuous at point and tends to . (cmptdst 25671 in the Hausdorff case.) Bourbaki TG I.50 cor. 2. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 8-Aug-2015.)

Theoremexopcopn 25675* For every neighborhood of in a product topology, there exist two open sets and of the component topologies so that is an open neighborhood of and a part of . (Use opelxp 4735 to have and .) (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremprdnei 25676 The product of two neighborhoods is a neighborhood. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 10-Jun-2014.)

Theoremlimptlimpr2lem1 25677 Lemma for limptlimpr 25679. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 9-Aug-2015.)

Theoremlimptlimpr2lem2 25678 Lemma for limptlimpr 25679. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 9-Aug-2015.)

Theoremlimptlimpr 25679 A limit in a product topology exists iff the limits of the projections exist. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 9-Aug-2015.)

Theoremflfnei2 25680* The property of being a limit point of a function in terms of filter and of preimage of a neighborhood. (Contributed by FL, 13-Dec-2013.) (Revised by Mario Carneiro, 8-Aug-2015.)

Syntaxcflimfrs 25681 Extend the definition of a class to include the limit of a function relatively to a subspace.

Definitiondf-flimfrs 25682* Gives the limits of a function at a point relatively to a subspace of a topology . ( The condition ensures the traces of the neighborhoods of over is a filter ( see trnei 17603). The set can't be empty since its closure is not empty ( see cldifemp 25627). Experimental. (Contributed by FL, 15-Sep-2013.)
t

Theoremislimrs 25683 The limits of at point when one only considers the traces of the neighborhoods of over . is a function whose domain is . The point must belong to (see also the comments under df-flimfrs 25682) . (Contributed by FL, 15-Sep-2013.)
t

Theoremislimrs3 25684 The limits of at point relatively to is a limit of at point relatively to . The opposite direction doesn't hold. (Contributed by FL, 13-Dec-2013.)
t

Theoremislimrs4 25685 The limits of at point relatively to is a limit of at point relatively to . (Contributed by FL, 13-Dec-2013.)
t

Syntaxcisopt 25686 Extend class notation to include isolated points.

Definitiondf-islpt 25687* Definition of an isolated point. Experimental. (Contributed by FL, 16-Sep-2013.)

18.13.33  Uniform spaces

Syntaxcunifsp 25688 Extend class notation with the class of all uniform spaces.

Definitiondf-unifsp 25689* Definition of a uniform space. Bourbaki TG II.1 def. 1. A uniform structure is used to give a generalization of the idea of Cauchy's sequence. We consider the space is equipped with the topology induced by the uniform structure. (Contributed by FL, 29-May-2014.)

18.13.34  Separated spaces: T0, T1, T2 (Hausdorff) ...

Theoremhst1 25690 A Hausdorff space is a T1 space. (Contributed by FL, 18-Jun-2007.)

Theoremdtt1 25691 A discrete topology is T1. Morris, Topology without tears. (Contributed by FL, 8-Jun-2007.)

18.13.35  Compactness

Theoremindcomp 25692 The indiscrete topology is compact. (Contributed by FL, 2-Aug-2009.)

Theoremtopunfincomp 25693 A topology whose underlying set is finite is compact. (Contributed by FL, 22-Dec-2008.)

Theoremstfincomp 25694 The subspace topology induced by a finite part of the underlying set of a topology is compact. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
t

Theorembwt2 25695* The glorious Bolzano-Weierstrass theorem. Certainly the first general topology theorem ever proved. In his course Weierstrass called it a lemma. He certainly didn't know how famous this theorem would be. He used an euclidian space instead of a general compact space. And he was not conscious of the Heine-Borel property. Cantor was one of his students. He used the concept of neighborhood and limit point invented by his master when he studied the linear point sets and the rest of the general topology followed from that. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)

18.13.36  Connectedness

Theoremsingempcon 25696 The singleton of the empty set is a connected topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremusinuniopb 25697 If a topology is connected, its underlying set can't be partitioned into two non-empty non-overlapping open sets. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 8-Apr-2015.)

Syntaxcopfn 25698 Extend class notation with an operator that derives an operation on functions from an operation on the elements of the common range of those functions.

Definitiondf-opfn 25699* Multiplication or addition of two functions and derived from the operation on the elements of the common range of and . The functions and must also have the same domain . (Contributed by FL, 15-Oct-2012.)

18.13.37  Topological fields

Syntaxctopfld 25700 Extend class notation to include TopFld.

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32776
 Copyright terms: Public domain < Previous  Next >