Home Metamath Proof ExplorerTheorem List (p. 175 of 325) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22374) Hilbert Space Explorer (22375-23897) Users' Mathboxes (23898-32447)

Theorem List for Metamath Proof Explorer - 17401-17500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremordthauslem 17401* Lemma for ordthaus 17402. (Contributed by Mario Carneiro, 13-Sep-2015.)
ordTop ordTop

Theoremordthaus 17402 The order topology of a total order is Hausdorff. (Contributed by Mario Carneiro, 13-Sep-2015.)
ordTop

11.1.11  Compactness

Syntaxccmp 17403 Extend class notation with the class of all compact spaces.

Definitiondf-cmp 17404* Definition of a compact topology. A topology is compact iff any open covering of its underlying set contains a finite sub-covering (Heine-Borel property). Definition C''' of [BourbakiTop1] p. I.59. Note: Bourbaki uses the term quasi-compact topology but it is not the modern usage (which we follow). (Contributed by FL, 22-Dec-2008.)

Theoremiscmp 17405* The predicate "is a compact topology". (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremcmpcov 17406* An open cover of a compact topology has a finite subcover. (Contributed by Jeff Hankins, 29-Jun-2009.)

Theoremcmpcov2 17407* Rewrite cmpcov 17406 for the cover . (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremcmpcovf 17408* Combine cmpcov 17406 with ac6sfi 7310 to show the existence of a function that indexes the elements that are generating the open cover. (Contributed by Mario Carneiro, 14-Sep-2014.)

Theoremcncmp 17409 Compactness is respected by a continuous onto map. (Contributed by Jeff Hankins, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)

Theoremfincmp 17410 A finite topology is compact. (Contributed by FL, 22-Dec-2008.)

Theorem0cmp 17411 The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.)

Theoremcmptop 17412 A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)

Theoremrncmp 17413 The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)
t

Theoremimacmp 17414 The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 18-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
t t

Theoremdiscmp 17415 A discrete topology is compact iff the base set is finite. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremcmpsublem 17416* Lemma for cmpsub 17417. (Contributed by Jeff Hankins, 28-Jun-2009.)
t t t

Theoremcmpsub 17417* Two equivalent ways of describing a compact subset of a topological space. Inspired by Sue E. Goodman's Beginning Topology. (Contributed by Jeff Hankins, 22-Jun-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremtgcmp 17418* A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 18029, which further specializes to subbases, assuming the ultrafilter lemma.) (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremcmpcld 17419 A closed subset of a compact space is compact. (Contributed by Jeff Hankins, 29-Jun-2009.)
t

Theoremuncmp 17420 The union of two compact sets is compact. (Contributed by Jeff Hankins, 30-Jan-2010.)
t t

Theoremfiuncmp 17421* A finite union of compact sets is compact. (Contributed by Mario Carneiro, 19-Mar-2015.)
t t

Theoremsscmp 17422 A subset of a compact topology (i.e. a coarser topology) is compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
TopOn

Theoremhauscmplem 17423* Lemma for hauscmp 17424. (Contributed by Mario Carneiro, 27-Nov-2013.)
t

Theoremhauscmp 17424 A compact subspace of a T2 space is closed. (Contributed by Jeff Hankins, 16-Jan-2010.) (Proof shortened by Mario Carneiro, 14-Dec-2013.)
t

Theoremcmpfi 17425* If a topology is compact and a collection of closed sets has the finite intersection property, its intersection is nonempty. (Contributed by Jeff Hankins, 25-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Theoremcmpfii 17426 In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)

11.1.12  Connectedness

Syntaxccon 17427 Extend class notation with the class of all connected topologies.

Definitiondf-con 17428 Topologies are connected when only and are both open and closed. (Contributed by FL, 17-Nov-2008.)

Theoremiscon 17429 The predicate is a connected topology . (Contributed by FL, 17-Nov-2008.)

Theoremiscon2 17430 The predicate is a connected topology . (Contributed by Mario Carneiro, 10-Mar-2015.)

Theoremconclo 17431 The only nonempty clopen set of a connected topology is the whole space. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theoremconndisj 17432 If a topology is connected, its underlying set can't be partitioned into two non-empty non-overlapping open sets. (Contributed by FL, 16-Nov-2008.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)

Theoremcontop 17433 A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.)

Theoremindiscon 17434 The indiscrete topology (or trivial topology) on any set is connected. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremdfcon2 17435* An alternate definition of connectedness. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
TopOn

Theoremconsuba 17436* Connectedness for a subspace. See connsub 17437. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
TopOn t

Theoremconnsub 17437* Two equivalent ways of saying that a subspace topology is connected. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
TopOn t

Theoremcnconn 17438 Connectedness is respected by a continuous onto map. (Contributed by Jeff Hankins, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)

Theoremnconsubb 17439 Disconnectedness for a subspace. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
TopOn                                                        t

Theoremconsubclo 17440 If a clopen set meets a connected subspace, it must contain the entire subspace. (Contributed by Mario Carneiro, 10-Mar-2015.)
t

Theoremconima 17441 The image of a connected set is connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
t        t

Theoremconcn 17442 A continuous function from a connected topology with one point in a clopen set must lie entirely within the set. (Contributed by Mario Carneiro, 16-Feb-2015.)

Theoremiunconlem 17443* Lemma for iuncon 17444. (Contributed by Mario Carneiro, 11-Jun-2014.)
TopOn                     t

Theoremiuncon 17444* The indexed union of connected overlapping subspaces sharing a common point is connected. (Contributed by Mario Carneiro, 11-Jun-2014.)
TopOn                     t        t

Theoremuncon 17445 The union of two connected overlapping subspaces is connected. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 11-Jun-2014.)
TopOn t t t

Theoremclscon 17446 The closure of a connected set is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)
TopOn t t

Theoremconcompid 17447* The connected component containing contains . (Contributed by Mario Carneiro, 19-Mar-2015.)
t        TopOn

Theoremconcompcon 17448* The connected component containing is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)
t        TopOn t

Theoremconcompss 17449* The connected component containing is a superset of any other connected set containing . (Contributed by Mario Carneiro, 19-Mar-2015.)
t        t

Theoremconcompcld 17450* The connected component containing is a closed set. (Contributed by Mario Carneiro, 19-Mar-2015.)
t        TopOn

Theoremconcompclo 17451* The connected component containing is a subset of any clopen set containing . (Contributed by Mario Carneiro, 20-Sep-2015.)
t        TopOn

Theoremt1conperf 17452 A connected T1 space is perfect, unless it is the topology of a singleton. (Contributed by Mario Carneiro, 26-Dec-2016.)
Perf

11.1.13  First- and second-countability

Syntaxc1stc 17453 Extend class definition to include the class of all first-countable topologies.

Syntaxc2ndc 17454 Extend class definition to include the class of all second-countable topologies.

Definitiondf-1stc 17455* Define the class of all first-countable topologies. (Contributed by Jeff Hankins, 22-Aug-2009.)

Definitiondf-2ndc 17456* Define the class of all second-countable topologies. (Contributed by Jeff Hankins, 17-Jan-2010.)

Theoremis1stc 17457* The predicate "is a first-countable topology." This can be described as "every point has a countable local basis" - that is, every point has a countable collection of open sets containing it such that every open set containing the point has an open set from this collection as a subset. (Contributed by Jeff Hankins, 22-Aug-2009.)

Theoremis1stc2 17458* An equivalent way of saying "is a first-countable topology." (Contributed by Jeff Hankins, 22-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)

Theorem1stctop 17459 A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)

Theorem1stcclb 17460* A property of points in a first-countable topology. (Contributed by Jeff Hankins, 22-Aug-2009.)

Theorem1stcfb 17461* For any point in a first-countable topology, there is a function enumerating neighborhoods of which is decreasing and forms a local base. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremis2ndc 17462* The property of being second-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.)

Theorem2ndctop 17463 A second-countable topology is a topology. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.)

Theorem2ndci 17464 A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theorem2ndcsb 17465* Having a countable subbase is a sufficient condition for second-countability. (Contributed by Jeff Hankins, 17-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theorem2ndcredom 17466 A second-countable space has at most the cardinality of the continuum. (Contributed by Mario Carneiro, 9-Apr-2015.)

Theorem2ndc1stc 17467 A second-countable space is first-countable. (Contributed by Jeff Hankins, 17-Jan-2010.)

Theorem1stcrestlem 17468* Lemma for 1stcrest 17469. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theorem1stcrest 17469 A subspace of a first-countable space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
t

Theorem2ndcrest 17470 A subspace of a second-countable space is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
t

Theorem2ndcctbss 17471* If a topology is second-countable, every base has a countable subset which is a base. Exercise 16B2 in Willard. (Contributed by Jeff Hankins, 28-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theorem2ndcdisj 17472* Any disjoint family of open sets in a second-countable space is countable. (The sets are required to be nonempty because otherwise there could be many empty sets in the family.) (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by Mario Carneiro, 9-Apr-2015.) (Revised by NM, 17-Jun-2017.)

Theorem2ndcdisj2 17473* Any disjoint collection of open sets in a second-countable space is countable. (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by Mario Carneiro, 9-Apr-2015.) (Revised by NM, 17-Jun-2017.)

Theorem2ndcomap 17474* A surjective continuous open map maps second-countable spaces to second-countable spaces. (Contributed by Mario Carneiro, 9-Apr-2015.)

Theorem2ndcsep 17475* A second-countable topology is separable, which is to say it contains a countable dense subset. (Contributed by Mario Carneiro, 13-Apr-2015.)

Theoremdis2ndc 17476 A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)

Theorem1stcelcls 17477* A point belongs to the closure of a subset iff there is a sequence in the subset converging to it. Theorem 1.4-6(a) of [Kreyszig] p. 30. This proof uses countable choice ax-cc 8271. A space satisfying the conclusion of this theorem is called a sequential space, so the theorem can also be stated as "every first-countable space is a sequential space". (Contributed by Mario Carneiro, 21-Mar-2015.)

Theorem1stccnp 17478* A mapping is continuous at in a first-countable space iff it is sequentially continuous at , meaning that the image under of every sequence converging at converges to . This proof uses ax-cc 8271, but only via 1stcelcls 17477, so it could be refactored into a proof that continuity and sequential continuity are the same in sequential spaces. (Contributed by Mario Carneiro, 7-Sep-2015.)
TopOn       TopOn

Theorem1stccn 17479* A mapping , where is first-countable, is continuous iff it is sequentially continuous, meaning that for any sequence converging to , its image under converges to . (Contributed by Mario Carneiro, 7-Sep-2015.)
TopOn       TopOn

11.1.14  Local topological properties

Syntaxclly 17480 Extend class notation with the "locally " predicate of a topological space.
Locally

Syntaxcnlly 17481 Extend class notation with the "N-locally " predicate of a topological space.
𝑛Locally

Definitiondf-lly 17482* Define a space that is locally , where is a topological property like "compact", "connected", or "path-connected". A topological space is locally if every neighborhood of a point contains an open sub-neighborhood that is in the subspace topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally t

Definitiondf-nlly 17483* Define a space that is n-locally , where is a topological property like "compact", "connected", or "path-connected". A topological space is n-locally if every neighborhood of a point contains a sub-neighborhood that is in the subspace topology.

The terminology "n-locally", where 'n' stands for "neighborhood", is not standard, although this is sometimes called "weakly locally ". The reason for the distinction is that some notions only make sense for arbitrary neighborhoods (such as "locally compact", which is actually 𝑛Locally in our teminology - open compact sets are not very useful), while others such as "locally connected" are strictly weaker notions if the neighborhoods are not required to be open. (Contributed by Mario Carneiro, 2-Mar-2015.)

𝑛Locally t

Theoremislly 17484* The property of being a locally topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally t

Theoremisnlly 17485* The property of being an n-locally topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally t

Theoremllyeq 17486 Equality theorem for the Locally predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally Locally

Theoremnllyeq 17487 Equality theorem for the Locally predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally 𝑛Locally

Theoremllytop 17488 A locally space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally

Theoremnllytop 17489 A locally space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally

Theoremllyi 17490* The property of a locally topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally t

Theoremnllyi 17491* The property of an n-locally topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally t

Theoremnlly2i 17492* Eliminate the neighborhood symbol from nllyi 17491. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally t

Theoremllynlly 17493 A locally space is n-locally : the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally 𝑛Locally

Theoremllyssnlly 17494 A locally space is n-locally . (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally 𝑛Locally

Theoremllyss 17495 The "locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Locally Locally

Theoremnllyss 17496 The "n-locally" predicate respects inclusion. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝑛Locally 𝑛Locally

Theoremsubislly 17497* The property of a subspace being locally . (Contributed by Mario Carneiro, 10-Mar-2015.)
t Locally t

Theoremrestnlly 17498* If the property passes to open subspaces, then a space is n-locally iff it is locally . (Contributed by Mario Carneiro, 2-Mar-2015.)
t        𝑛Locally Locally

Theoremrestlly 17499* If the property passes to open subspaces, then a space which is is also locally . (Contributed by Mario Carneiro, 2-Mar-2015.)
t               Locally

Theoremislly2 17500* An alternative expression for Locally when passes to open subspaces: A space is locally if every point is contained in an open neighborhood with property . (Contributed by Mario Carneiro, 2-Mar-2015.)
t               Locally t

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 32401-32447
 Copyright terms: Public domain < Previous  Next >