Home Metamath Proof ExplorerTheorem List (p. 171 of 327) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22413) Hilbert Space Explorer (22414-23936) Users' Mathboxes (23937-32699)

Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremistpsi 17001 Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)

Theoremeltpsg 17002 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet        TopOn

Theoremeltpsi 17003 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopSet

11.1.2  TopBases for topologies

Theoremisbasisg 17004* Express the predicate " is a basis for a topology." (Contributed by NM, 17-Jul-2006.)

Theoremisbasis2g 17005* Express the predicate " is a basis for a topology." (Contributed by NM, 17-Jul-2006.)

Theoremisbasis3g 17006* Express the predicate " is a basis for a topology." Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.)

Theorembasis1 17007 Property of a basis. (Contributed by NM, 16-Jul-2006.)

Theorembasis2 17008* Property of a basis. (Contributed by NM, 17-Jul-2006.)

Theoremfiinbas 17009* If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theorembasdif0 17010 A basis is not affected by the addition or removal of the empty set. (Contributed by Mario Carneiro, 28-Aug-2015.)

Theorembaspartn 17011* A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremtgval 17012* The topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremtgval2 17013* Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 17026) that is indeed a topology (on ; see unitg 17024). (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremeltg 17014 Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremeltg2 17015* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremeltg2b 17016* Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremeltg4i 17017 An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremeltg3i 17018 The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremeltg3 17019* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)

Theoremtgval3 17020* Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)

Theoremtg1 17021 Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)

Theoremtg2 17022* Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)

Theorembastg 17023 A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremunitg 17024 The topology generated by a basis is a topology on . Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.)

Theoremtgss 17025 Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)

Theoremtgcl 17026 Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)

Theoremtgclb 17027 The property tgcl 17026 can be reversed: if the topology generated by is actually a topology, then must be a topological basis. This yields an alternative definition of . (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtgtopon 17028 A basis generates a topology on . (Contributed by Mario Carneiro, 14-Aug-2015.)
TopOn

Theoremtopbas 17029 A topology is its own basis. (Contributed by NM, 17-Jul-2006.)

Theoremtgtop 17030 A topology is its own basis. (Contributed by NM, 18-Jul-2006.)

Theoremeltop 17031 Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)

Theoremeltop2 17032* Membership in a topology. (Contributed by NM, 19-Jul-2006.)

Theoremeltop3 17033* Membership in a topology. (Contributed by NM, 19-Jul-2006.)

Theoremfibas 17034 A collection of finite intersections is a basis. The initial set is a subbasis for the topology. (Contributed by Jeff Hankins, 25-Aug-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Theoremtgdom 17035 A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)

Theoremtgiun 17036* The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtgidm 17037 The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)

Theorembastop 17038 Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.)

Theoremtgtop11 17039 The topology generation function is one-to-one when applied to completed topologies. (Contributed by NM, 18-Jul-2006.)

Theorem0top 17040 The singleton of the empty set is the only topology possible for an empty underlying set. (Contributed by NM, 9-Sep-2006.)

Theoremen1top 17041 is the only topology with one element. (Contributed by FL, 18-Aug-2008.)

Theoremen2top 17042 If a topology has two elements, it is the indiscrete topology. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
TopOn

Theoremtgss3 17043 A criterion for determining whether one topology is finer than another. Lemma 2.2 of [Munkres] p. 80 using abbreviations. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Theoremtgss2 17044* A criterion for determining whether one topology is finer than another, based on a comparison of their bases. Lemma 2.2 of [Munkres] p. 80. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Theorembasgen 17045 Given a topology , show that a subset satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81 using abbreviations. (Contributed by NM, 22-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)

Theorembasgen2 17046* Given a topology , show that a subset satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Theorem2basgen 17047 Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 2-Sep-2015.)

Theoremtgfiss 17048 If a subbase is included into a topology, so is the generated topology. (Contributed by FL, 20-Apr-2012.) (Proof shortened by Mario Carneiro, 10-Jan-2015.)

Theoremtgdif0 17049 A generated topology is not affected by the addition or removal of the empty set from the base. (Contributed by Mario Carneiro, 28-Aug-2015.)

Theorembastop1 17050* A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom " " to express " is a basis for topology ," since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Theorembastop2 17051* A version of bastop1 17050 that doesn't have in the antecedent. (Contributed by NM, 3-Feb-2008.)

11.1.3  Examples of topologies

Theoremdistop 17052 The discrete topology on a set . Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 17-Jul-2006.) (Revised by Mario Carneiro, 19-Mar-2015.)

Theoremdistopon 17053 The discrete topology on a set , with base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremsn0topon 17054 The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremsn0top 17055 The singleton of the empty set is a topology. (Contributed by Stefan Allan, 3-Mar-2006.) (Proof shortened by Mario Carneiro, 13-Aug-2015.)

Theoremindislem 17056 A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremindistopon 17057 The indiscrete topology on a set . Part of Example 2 in [Munkres] p. 77. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremindistop 17058 The indiscrete topology on a set . Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 16-Jul-2006.) (Revised by Stefan Allan, 6-Nov-2008.) (Revised by Mario Carneiro, 13-Aug-2015.)

Theoremindisuni 17059 The base set of the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremfctop 17060* The finite complement topology on a set . Example 3 in [Munkres] p. 77. (Contributed by FL, 15-Aug-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremfctop2 17061* The finite complement topology on a set . Example 3 in [Munkres] p. 77. (This version of fctop 17060 requires the Axiom of Infinity.) (Contributed by FL, 20-Aug-2006.)
TopOn

Theoremcctop 17062* The countable complement topology on a set . Example 4 in [Munkres] p. 77. (Contributed by FL, 23-Aug-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremppttop 17063* The particular point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
TopOn

Theorempptbas 17064* The particular point topology is generated by a basis consisting of pairs for each . (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremepttop 17065* The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
TopOn

Theoremindistpsx 17066 The indiscrete topology on a set expressed as a topological space, using explicit structure component references. Compare with indistps 17067 and indistps2 17068. The advantage of this version is that the actual function for the structure is evident, and df-ndx 13464 is not needed, nor any other special definition outside of basic set theory. The disadvantage is that if the indices of the component definitions df-base 13466 and df-tset 13540 are changed in the future, this theorem will also have to be changed. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use indistps 17067 instead. (New usage is discouraged.) (Contributed by FL, 19-Jul-2006.)

Theoremindistps 17067 The indiscrete topology on a set expressed as a topological space, using implicit structure indices. The advantage of this version over indistpsx 17066 is that it is independent of the indices of the component definitions df-base 13466 and df-tset 13540, and if they are changed in the future, this theorem will not be affected. The advantage over indistps2 17068 is that it is easy to eliminate the hypotheses with eqid 2435 and vtoclg 3003 to result in a closed theorem. Theorems indistpsALT 17069 and indistps2ALT 17070 show that the two forms can be derived from each other. (Contributed by FL, 19-Jul-2006.)
TopSet

Theoremindistps2 17068 The indiscrete topology on a set expressed as a topological space, using direct component assignments. Compare with indistps 17067. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 17069 and indistps2ALT 17070 show that the two forms can be derived from each other. (Contributed by NM, 24-Oct-2012.)

TheoremindistpsALT 17069 The indiscrete topology on a set expressed as a topological space. Here we show how to derive the structural version indistps 17067 from the direct component assignment version indistps2 17068. (Contributed by NM, 24-Oct-2012.) (Proof modification is discouraged.)
TopSet

Theoremindistps2ALT 17070 The indiscrete topology on a set expressed as a topological space, using direct component assignments. Here we show how to derive the direct component assignment version indistps2 17068 from the structural version indistps 17067. (Contributed by NM, 24-Oct-2012.) (Proof modification is discouraged.)

Theoremdistps 17071 The discrete topology on a set expressed as a topological space. (Contributed by FL, 20-Aug-2006.)
TopSet

11.1.4  Closure and interior

Syntaxccld 17072 Extend class notation with the set of closed sets of a topology.

Syntaxcnt 17073 Extend class notation with interior of a subset of a topology base set.

Syntaxccl 17074 Extend class notation with closure of a subset of a topology base set.

Definitiondf-cld 17075* Define a function on topologies whose value is the set of closed sets of the topology. (Contributed by NM, 2-Oct-2006.)

Definitiondf-ntr 17076* Define a function on topologies whose value is the interior function on the subsets of the base set. See ntrval 17092. (Contributed by NM, 10-Sep-2006.)

Definitiondf-cls 17077* Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 17093. (Contributed by NM, 3-Oct-2006.)

Theoremfncld 17078 The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremcldval 17079* The set of closed sets of a topology. (Note that the set of open sets is just the topology itself, so we don't have a separate definition.) (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremntrfval 17080* The interior function on the subsets of a topology's base set. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremclsfval 17081* The closure function on the subsets of a topology's base set. (Contributed by NM, 3-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremcldrcl 17082 Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremiscld 17083 The predicate " is a closed set." (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremiscld2 17084 A subset of the underlying set of a topology is closed iff its complement is open. (Contributed by NM, 4-Oct-2006.)

Theoremcldss 17085 A closed set is a subset of the underlying set of a topology. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)

Theoremcldss2 17086 The set of closed sets is contained in the powerset of the base. (Contributed by Mario Carneiro, 6-Jan-2014.)

Theoremcldopn 17087 The complement of a closed set is open. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)

Theoremisopn2 17088 A subset of the underlying set of a topology is open iff its complement is closed. (Contributed by NM, 4-Oct-2006.)

Theoremopncld 17089 The complement of an open set is closed. (Contributed by NM, 6-Oct-2006.)

Theoremdifopn 17090 The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.)

Theoremtopcld 17091 The underlying set of a topology is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 3-Oct-2006.)

Theoremntrval 17092 The interior of a subset of a topology's base set is the union of all the open sets it includes. Definition of interior of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremclsval 17093* The closure of a subset of a topology's base set is the intersection of all the closed sets that include it. Definition of closure of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theorem0cld 17094 The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 4-Oct-2006.)

Theoremiincld 17095* The indexed intersection of a collection of closed sets is closed. Theorem 6.1(2) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.) (Revised by Mario Carneiro, 3-Sep-2015.)

Theoremintcld 17096 The intersection of a set of closed sets is closed. (Contributed by NM, 5-Oct-2006.)

Theoremuncld 17097 The union of two closed sets is closed. Equivalent to Theorem 6.1(3) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.)

Theoremcldcls 17098 A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)

Theoremincld 17099 The intersection of two closed sets is closed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)

Theoremriincld 17100* An indexed relative intersection of closed sets is closed. (Contributed by Stefan O'Rear, 22-Feb-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 30801-30900 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 32601-32699
 Copyright terms: Public domain < Previous  Next >