Home Metamath Proof ExplorerTheorem List (p. 170 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 - 16901-17000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempnrmtop 16901 A perfectly normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theorempnrmcld 16902* A closed set in a perfectly normal space is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theorempnrmopn 16903* An open set in a perfectly normal space is a countable union of closed sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theoremist0-2 16904* The predicate "is a T0 space". (Contributed by Mario Carneiro, 24-Aug-2015.)
TopOn

Theoremist0-3 16905* The predicate "is a T0 space," expressed in more familiar terms. (Contributed by Jeff Hankins, 1-Feb-2010.)
TopOn

Theoremcnt0 16906 The preimage of a T0 topology under an injective map is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremist1-2 16907* An alternate characterization of T1 spaces. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
TopOn

Theoremt1t0 16908 A T1 space is a T0 space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremist1-3 16909* A space is T1 iff every point is the only point in the intersection of all open sets containing that point. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
TopOn

Theoremcnt1 16910 The preimage of a T1 topology under an injective map is T1. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremishaus2 16911* Express the predicate " is a Hausdorff space." (Contributed by NM, 8-Mar-2007.)
TopOn

Theoremhaust1 16912 A Hausdorff space is a T1 space. (Contributed by FL, 11-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)

Theoremhausnei2 16913* The Hausdorff condition still holds if one considers general neighborhoods instead of open sets. (Contributed by Jeff Hankins, 5-Sep-2009.)
TopOn

Theoremcnhaus 16914 The preimage of a Hausdorff topology under an injective map is Hausdorff. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremnrmsep3 16915* In a normal space, given a closed set inside an open set , there is an open set such that . (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremnrmsep2 16916* In a normal space, any two disjoint closed sets have the property that each one is a subset of an open set whose closure is disjoint from the other. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremnrmsep 16917* In a normal space, disjoint closed sets are separated by open sets. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremisnrm2 16918* An alternate characterization of normality. This is the important property in the proof of Urysohn's lemma. (Contributed by Jeff Hankins, 1-Feb-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)

Theoremisnrm3 16919* A topological space is normal iff any two disjoint closed sets are separated by open sets. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremcnrmi 16920 A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm t

Theoremcnrmnrm 16921 A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm

Theoremrestcnrm 16922 A subspace of a completely normal space is completely normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm t CNrm

Theoremresthauslem 16923 Lemma for resthaus 16928 and similar theorems. If the topological property is preserved under injective preimages, then property passes to subspaces. (Contributed by Mario Carneiro, 25-Aug-2015.)
t t        t

Theoremlpcls 16924 The limit points of the closure of a subset are the same as the limit points of the set in a T1 space. (Contributed by Mario Carneiro, 26-Dec-2016.)

Theoremperfcls 16925 A subset of a perfect space is perfect iff its closure is perfect (and the closure is an actual perfect set, since it is both closed and perfect in the subspace topology). (Contributed by Mario Carneiro, 26-Dec-2016.)
t Perf t Perf

Theoremrestt0 16926 A subspace of a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
t

Theoremrestt1 16927 A subspace of a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
t

Theoremresthaus 16928 A subspace of a Hausdorff topology is Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
t

Theoremt1sep2 16929* Any two points in a T1 space which have no separation are equal. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremt1sep 16930* Any two distinct points in a T1 space are separated by an open set. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremsncld 16931 A singleton is closed in a Hausdorff space. (Contributed by NM, 5-Mar-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremsshauslem 16932 Lemma for sshaus 16935 and similar theorems. If the topological property is preserved under injective preimages, then a topology finer than one with property also has property . (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremsst0 16933 A topology finer than a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremsst1 16934 A topology finer than a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremsshaus 16935 A topology finer than a Hausdorff topology is Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015.)
TopOn

Theoremregsep2 16936* In a regular space, a closed set is separated by open sets from a point not in it. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)

Theoremisreg2 16937* A topological space is regular if any closed set is separated from any point not in it by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)
TopOn

Theoremdnsconst 16938 If a continuous mapping to a T1 space is constant on a dense subset, it is constant on the entire space. Note that means " is dense in " and means " is constant on " (see funconstss 5495). (Contributed by NM, 15-Mar-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)

Theoremordtt1 16939 The order topology is T1 for any poset. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremlmmo 16940 A sequence in a Hausdorff space converges to at most one limit. Part of Lemma 1.4-2(a) of [Kreyszig] p. 26. (Contributed by NM, 31-Jan-2008.) (Proof shortened by Mario Carneiro, 1-May-2014.)

Theoremlmfun 16941 The convergence relation is function-like in a Hausdorff space. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremdishaus 16942 A discrete topology is Hausdorff. Morris, Topology without tears, p.72, ex. 13. (Contributed by FL, 24-Jun-2007.) (Proof shortened by Mario Carneiro, 8-Apr-2015.)

Theoremordthauslem 16943* Lemma for ordthaus 16944. (Contributed by Mario Carneiro, 13-Sep-2015.)
ordTop ordTop

Theoremordthaus 16944 The order topology of a total order is Hausdorff. (Contributed by Mario Carneiro, 13-Sep-2015.)
ordTop

11.1.11  Compactness

Syntaxccmp 16945 Extend class notation with the class of all compact spaces.

Definitiondf-cmp 16946* 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 16947* The predicate "is a compact topology". (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremcmpcov 16948* An open cover of a compact topology has a finite subcover. (Contributed by Jeff Hankins, 29-Jun-2009.)

Theoremcmpcov2 16949* Rewrite cmpcov 16948 for the cover . (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremcmpcovf 16950* Combine cmpcov 16948 with ac6sfi 6986 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 16951 Compactness is respected by a continuous onto map. (Contributed by Jeff Hankins, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)

Theoremfincmp 16952 A finite topology is compact. (Contributed by FL, 22-Dec-2008.)

Theorem0cmp 16953 The singleton of the empty set is compact. (Contributed by FL, 2-Aug-2009.)

Theoremcmptop 16954 A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)

Theoremrncmp 16955 The image of a compact set under a continuous function is compact. (Contributed by Mario Carneiro, 21-Mar-2015.)
t

Theoremimacmp 16956 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 16957 A discrete topology is compact iff the base set is finite. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremcmpsublem 16958* Lemma for cmpsub 16959. (Contributed by Jeff Hankins, 28-Jun-2009.)
t t t

Theoremcmpsub 16959* 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 16960* A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 17571, which further specializes to subbases, assuming the ultrafilter lemma.) (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremcmpcld 16961 A closed subset of a compact space is compact. (Contributed by Jeff Hankins, 29-Jun-2009.)
t

Theoremuncmp 16962 The union of two compact sets is compact. (Contributed by Jeff Hankins, 30-Jan-2010.)
t t

Theoremfiuncmp 16963* A finite union of compact sets is compact. (Contributed by Mario Carneiro, 19-Mar-2015.)
t t

Theoremsscmp 16964 A subset of a compact topology (i.e. a coarser topology) is compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
TopOn

Theoremhauscmplem 16965* Lemma for hauscmp 16966. (Contributed by Mario Carneiro, 27-Nov-2013.)
t

Theoremhauscmp 16966 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 16967* 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 16968 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 16969 Extend class notation with the class of all connected topologies.

Definitiondf-con 16970 Topologies are connected when only and are both open and closed. (Contributed by FL, 17-Nov-2008.)

Theoremiscon 16971 The predicate is a connected topology . (Contributed by FL, 17-Nov-2008.)

Theoremiscon2 16972 The predicate is a connected topology . (Contributed by Mario Carneiro, 10-Mar-2015.)

Theoremconclo 16973 The only nonempty clopen set of a connected topology is the whole space. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theoremconndisj 16974 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 16975 A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.)

Theoremindiscon 16976 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 16977* An alternate definition of connectedness. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
TopOn

Theoremconsuba 16978* Connectedness for a subspace. See connsub 16979. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
TopOn t

Theoremconnsub 16979* 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 16980 Connectedness is respected by a continuous onto map. (Contributed by Jeff Hankins, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)

Theoremnconsubb 16981 Disconnectedness for a subspace. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
TopOn                                                        t

Theoremconsubclo 16982 If a clopen set meets a connected subspace, it must contain the entire subspace. (Contributed by Mario Carneiro, 10-Mar-2015.)
t

Theoremconima 16983 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 16984 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 16985* Lemma for iuncon 16986. (Contributed by Mario Carneiro, 11-Jun-2014.)
TopOn                     t

Theoremiuncon 16986* The indexed union of connected overlapping subspaces sharing a common point is connected. (Contributed by Mario Carneiro, 11-Jun-2014.)
TopOn                     t        t

Theoremuncon 16987 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 16988 The closure of a connected set is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)
TopOn t t

Theoremconcompid 16989* The connected component containing contains . (Contributed by Mario Carneiro, 19-Mar-2015.)
t        TopOn

Theoremconcompcon 16990* The connected component containing is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)
t        TopOn t

Theoremconcompss 16991* The connected component containing is a superset of any other connected set containing . (Contributed by Mario Carneiro, 19-Mar-2015.)
t        t

Theoremconcompcld 16992* The connected component containing is a closed set. (Contributed by Mario Carneiro, 19-Mar-2015.)
t        TopOn

Theoremconcompclo 16993* The connected component containing is a subset of any clopen set containing . (Contributed by Mario Carneiro, 20-Sep-2015.)
t        TopOn

Theoremt1conperf 16994 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 16995 Extend class definition to include the class of all first-countable topologies.

Syntaxc2ndc 16996 Extend class definition to include the class of all second-countable topologies.

Definitiondf-1stc 16997* Define the class of all first-countable topologies. (Contributed by Jeff Hankins, 22-Aug-2009.)

Definitiondf-2ndc 16998* Define the class of all second-countable topologies. (Contributed by Jeff Hankins, 17-Jan-2010.)

Theoremis1stc 16999* 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 17000* An equivalent way of saying "is a first-countable topology." (Contributed by Jeff Hankins, 22-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-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 >