Home Metamath Proof ExplorerTheorem List (p. 173 of 309) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21328) Hilbert Space Explorer (21329-22851) Users' Mathboxes (22852-30843)

Theorem List for Metamath Proof Explorer - 17201-17300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcnmpt22f 17201* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmpt1res 17202* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.)
t        TopOn

Theoremcnmpt2res 17203* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
t        TopOn              t        TopOn

Theoremcnmptcom 17204* The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
TopOn       TopOn

Theoremcnmptkc 17205* The curried first projection function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn

Theoremcnmptkp 17206* The evaluation of the inner function in a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn

Theoremcnmptk1 17207* The composition of a curried function with a one-arg function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn       TopOn       TopOn

Theoremcnmpt1k 17208* The composition of a one-arg function with a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn       TopOn

Theoremcnmptkk 17209* The composition of two curried functions is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn       TopOn       𝑛Locally

Theoremxkofvcn 17210* Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 17182.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝑛Locally

Theoremcnmptk1p 17211* The evaluation of a curried function by a one-arg function is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn       𝑛Locally

Theoremcnmptk2 17212* The uncurrying of a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn       TopOn       TopOn       𝑛Locally

Theoremxkoinjcn 17213* Continuity of "injection", i.e. currying, as a function on continuous function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn TopOn

Theoremcnmpt2k 17214* The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn       TopOn

Theoremtxcon 17215 The topological product of two connected spaces is connected. (Contributed by Mario Carneiro, 29-Mar-2015.)

11.1.18  Quotient maps and quotient topology

Syntaxckq 17216 Extend class notation with the Kolmogorov quotient function.
KQ

Definitiondf-kq 17217* Define the Kolmogorov quotient. This is a function on topologies which maps a topology to its quotient under the topological distinguishability map, which takes a point to the set of open sets that contain it. Two points are mapped to the same image under this function iff they are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ qTop

Theoremqtopval 17218* Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremqtopval2 17219* Value of the quotient topology function when is a function on the base set. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremelqtop 17220 Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremqtopres 17221 The quotient topology is unaffected by restriction to the base set. This property makes it slightly more convenient to use, since we don't have to require that be a function with domain . (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop qTop

Theoremqtoptop2 17222 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremqtoptop 17223 The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremelqtop2 17224 Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
qTop

Theoremqtopuni 17225 The base set of the quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
qTop

Theoremelqtop3 17226 Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.)
TopOn qTop

Theoremqtoptopon 17227 The base set of the quotient topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn qTop TopOn

Theoremqtopid 17228 A quotient map a continuous function into its quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn qTop

Theoremidqtop 17229 The quotient topology induced by the identity function is the original topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
TopOn qTop

Theoremqtopcmplem 17230 Lemma for qtopcmp 17231 and qtopcon 17232. (Contributed by Mario Carneiro, 24-Mar-2015.)
qTop qTop qTop        qTop

Theoremqtopcmp 17231 A quotient of a compact space is compact. (Contributed by Mario Carneiro, 24-Mar-2015.)
qTop

Theoremqtopcon 17232 A quotient of a connected space is connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
qTop

Theoremqtopkgen 17233 A quotient of a compactly generated space is compactly generated. (Contributed by Mario Carneiro, 9-Apr-2015.)
𝑘Gen qTop 𝑘Gen

Theorembasqtop 17234 An injection maps bases to bases. (Contributed by Mario Carneiro, 27-Aug-2015.)
qTop

Theoremtgqtop 17235 An injection maps generated topologies to each other. (Contributed by Mario Carneiro, 27-Aug-2015.)
qTop qTop

Theoremqtopcld 17236 The property of being a closed set in the quotient topology. (Contributed by Mario Carneiro, 24-Mar-2015.)
TopOn qTop

Theoremqtopcn 17237 Universal property of a quotient map. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn TopOn qTop

Theoremqtopss 17238 A surjective continuous function from to induces a topology qTop on the base set of . This topology is in general finer than . Together with qtopid 17228, this implies that qTop is the finest topology making continuous, i.e. the final topology with respect to the family . (Contributed by Mario Carneiro, 24-Mar-2015.)
TopOn qTop

Theoremqtopeu 17239* Universal property of the quotient topology. If is a function from to which is equal on all equivalent elements under , then there is a unique continuous map such that , and we say that "passes to the quotient". (Contributed by Mario Carneiro, 24-Mar-2015.)
TopOn                            qTop

Theoremqtoprest 17240 If is a saturated open or closed set (where saturated means that for some ), then the restriction of the quotient map to is a quotient map. (Contributed by Mario Carneiro, 24-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn                                   qTop t t qTop

Theoremqtopomap 17241* If is a surjective continuous open map, then it is a quotient map. (An open map is a function that maps open sets to open sets.) (Contributed by Mario Carneiro, 24-Mar-2015.)
TopOn                            qTop

Theoremqtopcmap 17242* If is a surjective continuous closed map, then it is a quotient map. (A closed map is a function that maps closed sets to closed sets.) (Contributed by Mario Carneiro, 24-Mar-2015.)
TopOn                            qTop

Theoremimastopn 17243 The topology of an image structure. (Contributed by Mario Carneiro, 27-Aug-2015.)
s                                           qTop

Theoremimastps 17244 The image of a topological space under a function is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremdivstps 17245 A quotient structure is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
s

Theoremkqfval 17246* Value of the function appearing in df-kq 17217. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremkqfeq 17247* Two points in the Kolmogorov quotient are equal iff the original points are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremkqffn 17248* The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremkqval 17249* Value of the quotient topology function. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ qTop

Theoremkqtopon 17250* The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ TopOn

Theoremkqid 17251* The topological indistinguishability map is a continuous function into the Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremist0-4 17252* The topological indistinguishability map is injective iff the space is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqfvima 17253* When the image set is open, the quotient map satisfies a partial converse to fnfvima 5608, which is normally only true for injective functions. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqsat 17254* Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 17240). (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqdisj 17255* A version of imain 5185 for the topological indistinguishability map. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqcldsat 17256* Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 17240). (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremkqopn 17257* The topological indistinguishability map is an open map. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqcld 17258* The topological indistinguishability map is a closed map. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqt0lem 17259* Lemma for kqt0 17269. (Contributed by Mario Carneiro, 23-Mar-2015.)
TopOn KQ

Theoremisr0 17260* The property " is an R0 space". A space is R0 if any two topologically distinguishable points are separated (there is an open set containing each one and disjoint from the other). Or in contraposition, if every open set which contains also contains , so there is no separation, then and are members of the same open sets. We have chosen not to give this definition a name, because it turns out that a space is R0 if and only if its Kolmogorov quotient is T1, so that is what we prove here. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremr0cld 17261* The analogue of the T1 axiom (singletons are closed) for an R0 space. In an R0 space the set of all points topologically indistinguishable from is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremregr1lem 17262* Lemma for regr1 17273. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn                                   KQ KQ

Theoremregr1lem2 17263* A Kolmogorov quotient of a regular space is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqreglem1 17264* A Kolmogorov quotient of a regular space is regular. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqreglem2 17265* If the Kolmogorov quotient of a space is regular then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqnrmlem1 17266* A Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqnrmlem2 17267* If the Kolmogorov quotient of a space is normal then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremkqtop 17268 The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremkqt0 17269 The Kolmogorov quotient is T0 even if the original topology is not. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremkqf 17270 The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremr0sep 17271* The separation property of an R0 space. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn KQ

Theoremnrmr0reg 17272 A normal R0 space is also regular. These spaces are usually referred to as normal regular spaces. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremregr1 17273 A regular space is R1, which means that any two topologically distinct points can be separated by neighborhoods. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremkqreg 17274 The Kolmogorov quotient of a regular space is regular. By regr1 17273 it is also Hausdorff, so we can also say that a space is regular iff the Kolmogorov quotient is regular Hausdorff (T3). (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

Theoremkqnrm 17275 The Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
KQ

11.1.19  Homeomorphisms

Syntaxchmeo 17276 Extend class notation with the class of all homeomorphisms.

Syntaxchmph 17277 Extend class notation with the relation "is homeomorph to.".

Definitiondf-hmeo 17278* Function returning all the homeomorphisms from topology to topology . (Contributed by FL, 14-Feb-2007.)

Definitiondf-hmph 17279 Definition of the relation is homeomorph to . (Contributed by FL, 14-Feb-2007.)

Theoremhmeofn 17280 The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremhmeofval 17281* The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremishmeo 17282 The predicate F is a homeomorphism between topology and topology . Proposition of [BourbakiTop1] p. I.2. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremhmeocn 17283 A homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)

Theoremhmeocnvcn 17284 The converse of a homeomorphism is continuous. (Contributed by Mario Carneiro, 22-Aug-2015.)

Theoremhmeocnv 17285 The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremhmeof1o2 17286 A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theoremhmeof1o 17287 A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)

Theoremhmeoima 17288 The image of an open set by a homeomorphism is an open set. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremhmeoopn 17289 Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)

Theoremhmeocld 17290 Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)

Theoremhmeocls 17291 Homeomorphisms preserve closures. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremhmeontr 17292 Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremhmeoimaf1o 17293* The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)

Theoremhmeores 17294 The restriction of a homeomorphism is a homeomorphism. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
t t

Theoremhmeoco 17295 The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Theoremidhmeo 17296 The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
TopOn

Theoremhmeocnvb 17297 The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Theoremhmeoqtop 17298 A homeomorphism is a quotient map. (Contributed by Mario Carneiro, 25-Aug-2015.)
qTop

Theoremhmph 17299 Express the predicate is homeomorph to . (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)

Theoremhmphi 17300 If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.)

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 30801-30843
 Copyright terms: Public domain < Previous  Next >