Home Metamath Proof ExplorerTheorem List (p. 169 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 - 16801-16900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiscnp2 16801* The predicate " is a continuous function from topology to topology at point ." Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcntop1 16802 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcntop2 16803 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnptop1 16804 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnptop2 16805 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremiscnp3 16806* The predicate " is a continuous function from topology to topology at point ." (Contributed by NM, 15-May-2007.)
TopOn TopOn

Theoremcnprcl 16807 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcnf 16808 A continuous function is a mapping. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnpf 16809 A continuous function at point is a mapping. (Contributed by FL, 17-Nov-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnpcl 16810 The value of a continuous function from to at point belongs to the underlying set of topology . (Contributed by FL, 27-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnf2 16811 A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnpf2 16812 A continuous function at point is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnprcl2 16813 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremtgcn 16814* The contininuity predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn              TopOn

Theoremtgcnp 16815* The "continuous at a point" predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn              TopOn

Theoremsubbascn 16816* The contininuity predicate when the range is given by a subbasis for a topology. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
TopOn                     TopOn

Theoremssidcn 16817 The identity function is a continuous function from one topology to another topology on the same set iff the domain is finer than the codomain. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnpimaex 16818* Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)

Theoremidcn 16819 A restricted identity function is a continuous function. (Contributed by FL, 27-Dec-2006.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
TopOn

Theoremlmbr 16820* Express the binary relation "sequence converges to point " in a topological space. Definition 1.4-1 of [Kreyszig] p. 25. The condition allows us to use objects more general than sequences when convenient; see the comment in df-lm 16791. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmbr2 16821* Express the binary relation "sequence converges to point " in a metric space using an arbitrary set of upper integers. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmbrf 16822* Express the binary relation "sequence converges to point " in a metric space using an arbitrary set of upper integers. This version of lmbr2 16821 presupposes that is a function. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmconst 16823 A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmcvg 16824* Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremcnpnei 16825* A condition for continuity at a point in terms of neighborhoods. (Contributed by Jeff Hankins, 7-Sep-2009.)

Theoremcnima 16826 An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.)

Theoremcnco 16827 The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnpco 16828 The composition of two continuous functions at point is a continuous function at point . Proposition of [BourbakiTop1] p. I.9. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theoremcnclima 16829 A closed subset of the codomain of a continuous function has a closed preimage. (Contributed by NM, 15-Mar-2007.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremiscncl 16830* A definition of a continuous function using closed sets. Theorem 1 (d) of [BourbakiTop1] p. I.9. (Contributed by FL, 19-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcncls2i 16831 Property of the preimage of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcnntri 16832 Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcnclsi 16833 Property of the image of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcncls2 16834* Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
TopOn TopOn

Theoremcncls 16835* Continuity in terms of closure. (Contributed by Jeff Hankins, 1-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
TopOn TopOn

Theoremcnntr 16836* Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
TopOn TopOn

Theoremcnss1 16837 If the topology is finer than , then there are more continuous functions from than from . (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremcnss2 16838 If the topology is finer than , then there are fewer continuous functions into than into from some other space. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremcncnpi 16839 A continuous function is continuous at all points. One direction of Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)

Theoremcnsscnp 16840 The set of continuous functions is a subset of the set of continuous functions at a point. (Contributed by Raph Levien, 21-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcncnp 16841* A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 15-May-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcncnp2 16842* A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)

Theoremcnconst2 16843 A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
TopOn TopOn

Theoremcnconst 16844 A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)
TopOn TopOn

Theoremcnrest 16845 Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.)
t

Theoremcnrest2 16846 Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Hankins, 10-Jul-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
TopOn t

Theoremcnrest2r 16847 Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)
t

Theoremcnpresti 16848 One direction of cnprest 16849 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 1-May-2015.)
t

Theoremcnprest 16849 Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.)
t

Theoremcnprest2 16850 Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 21-Aug-2015.)
t

Theoremcndis 16851 Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremcnindis 16852 Every function is continuous when the codomain is indiscrete (trivial). (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremcnpdis 16853 If is an isolated point in (or equivalently, the singleton is open in ), then every function is continuous at . (Contributed by Mario Carneiro, 9-Sep-2015.)
TopOn TopOn

Theorempaste 16854 Pasting lemma. If and are closed sets in with , then any function whose restrictions to and are continuous is continuous on all of . (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
t        t

Theoremlmfpm 16855 If converges, then is a partial function. (Contributed by Mario Carneiro, 23-Dec-2013.)
TopOn

Theoremlmfss 16856 Inclusion of a function having a limit (used to ensure the limit relation is a set, under our definition). (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
TopOn

Theoremlmcl 16857 Closure of a limit. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
TopOn

Theoremlmss 16858 Limit on a subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
t

Theoremsslm 16859 A finer topology has fewer convergent sequences (but the sequences that do converge converge to the same value). (Contributed by Mario Carneiro, 15-Sep-2015.)
TopOn TopOn

Theoremlmres 16860 A function converges iff its restriction to an upper integers set converges. (Contributed by Mario Carneiro, 31-Dec-2013.)
TopOn

Theoremlmff 16861* If converges, there is some upper integer set on which is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)
TopOn

Theoremlmcls 16862* Any convergent sequence of points in a subset of a topological space converges to a point in the closure of the subset. (Contributed by Mario Carneiro, 30-Dec-2013.) (Revised by Mario Carneiro, 1-May-2014.)
TopOn

Theoremlmcld 16863* Any convergent sequence of points in a closed subset of a topological space converges to a point in the set. (Contributed by Mario Carneiro, 30-Dec-2013.)
TopOn

Theoremlmcnp 16864 The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.)

Theoremlmcn 16865 The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.)

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

Syntaxct0 16866 Extend class notation with the class of all T0 spaces.

Syntaxct1 16867 Extend class notation to include T1 spaces (also called Fréchet spaces).

Syntaxcha 16868 Extend class notation with the class of all Hausdorff spaces.

Syntaxcreg 16869 Extend class notation with the class of all regular topologies.

Syntaxcnrm 16870 Extend class notation with the class of all normal topologies.

Syntaxccnrm 16871 Extend class notation with the class of all completely normal topologies.
CNrm

Syntaxcpnrm 16872 Extend class notation with the class of all perfectly normal topologies.
PNrm

Definitiondf-t0 16873* Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2234): any two points which are members of the same open sets are equal, or in contraposition, for any two distinct points there is an open set which contains one point but not the other. This differs from T1 spaces (see ist1-2 16907) in that in a T1 space you can choose which point will be in the open set and which outside; in a T0 space you only know that one of the two points is in the set. (Contributed by Jeff Hankins, 1-Feb-2010.)

Definitiondf-t1 16874* The class of all T1 spaces, also called Fréchet spaces. Morris, Topology without tears, p. 30 ex. 3. (Contributed by FL, 18-Jun-2007.)

Definitiondf-haus 16875* Define the class of all Hausdorff spaces. A Hausdorff space is a topology in which distinct points have disjoint open neighborhoods. Definition of Hausdorff space in [Munkres] p. 98. (Contributed by NM, 8-Mar-2007.)

Definitiondf-reg 16876* Define regular spaces. A space is regular if a point and a closed set can be separated by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.)

Definitiondf-nrm 16877* Define normal spaces. A space is normal if disjoint closed sets can be separated by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.)

Definitiondf-cnrm 16878* Define completely normal spaces. A space is completely normal if all its subspaces are normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm t

Definitiondf-pnrm 16879* Define perfectly normal spaces. A space is perfectly normal if it is normal and every closed set is a G&delta; set, meaning that it is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theoremist0 16880* The predicate "is a T0 space." Every pair of distinct points is topologically distinguishable. For the way this definition is usually encountered, see ist0-3 16905. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremist1 16881* The predicate is T1. (Contributed by FL, 18-Jun-2007.)

Theoremishaus 16882* Express the predicate " is a Hausdorff space." (Contributed by NM, 8-Mar-2007.)

Theoremiscnrm 16883* The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm t

Theoremt0sep 16884* Any two topologically indistinguishable points in a T0 space are identical. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremt0dist 16885* Any two distinct points in a T0 space are topologically distinguishable. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremt1sncld 16886 In a T1 space, one-point sets are closed. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremt1ficld 16887 In a T1 space, finite sets are closed. (Contributed by Mario Carneiro, 25-Dec-2016.)

Theoremhausnei 16888* Neighborhood property of a Hausdorff space. (Contributed by NM, 8-Mar-2007.)

Theoremt0top 16889 A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremt1top 16890 A T1 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremhaustop 16891 A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.)

Theoremisreg 16892* The predicate "is a regular space." In a regular space, any open neighborhood has a closed subneighborhood. Note that some authors require the space to be Hausdorff (which would make it the same as T3), but we reserve the phrase "regular Hausdorff" for that as many topologists do. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)

Theoremregtop 16893 A regular space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremregsep 16894* In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremisnrm 16895* The predicate "is a normal space." Much like the case for regular spaces, normal does not imply Hausdorff or even regular. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremnrmtop 16896 A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Theoremcnrmtop 16897 A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
CNrm

Theoremiscnrm2 16898* The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
TopOn CNrm t

Theoremispnrm 16899* The property of being perfectly normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

Theorempnrmnrm 16900 A perfectly normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
PNrm

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 >