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Theorem List for Intuitionistic Logic Explorer - 12401-12500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmstri 12401 Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremxmstri3 12402 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremmstri3 12403 Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremmsrtri 12404 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremxmspropd 12405 Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremmspropd 12406 Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremsetsmsbasg 12407 The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
sSet TopSet

Theoremsetsmsdsg 12408 The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
sSet TopSet

Theoremsetsmstsetg 12409 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Revised by Jim Kingdon, 7-May-2023.)
sSet TopSet                      TopSet

Theoremmopni 12410* An open set of a metric space includes a ball around each of its points. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremmopni2 12411* An open set of a metric space includes a ball around each of its points. (Contributed by NM, 2-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremmopni3 12412* An open set of a metric space includes an arbitrarily small ball around each of its points. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremblssopn 12413 The balls of a metric space are open sets. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)

Theoremunimopn 12414 The union of a collection of open sets of a metric space is open. Theorem T2 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)

Theoremmopnin 12415 The intersection of two open sets of a metric space is open. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)

Theoremmopn0 12416 The empty set is an open set of a metric space. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.)

Theoremrnblopn 12417 A ball of a metric space is an open set. (Contributed by NM, 12-Sep-2006.)

Theoremblopn 12418 A ball of a metric space is an open set. (Contributed by NM, 9-Mar-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremneibl 12419* The neighborhoods around a point of a metric space are those subsets containing a ball around . Definition of neighborhood in [Kreyszig] p. 19. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)

Theoremblnei 12420 A ball around a point is a neighborhood of the point. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremblsscls2 12421* A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)

Theoremmetss 12422* Two ways of saying that metric generates a finer topology than metric . (Contributed by Mario Carneiro, 12-Nov-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremmetequiv 12423* Two ways of saying that two metrics generate the same topology. Two metrics satisfying the right-hand side are said to be (topologically) equivalent. (Contributed by Jeff Hankins, 21-Jun-2009.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremmetequiv2 12424* If there is a sequence of radii approaching zero for which the balls of both metrics coincide, then the generated topologies are equivalent. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremmetss2lem 12425* Lemma for metss2 12426. (Contributed by Mario Carneiro, 14-Sep-2015.)

Theoremmetss2 12426* If the metric is "strongly finer" than (meaning that there is a positive real constant such that ), then generates a finer topology. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they generate the same topology.) (Contributed by Mario Carneiro, 14-Sep-2015.)

Theoremcomet 12427* The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theorembdmetval 12428* Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
inf        inf

Theorembdxmet 12429* The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
inf

Theorembdmet 12430* The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
inf

Theorembdbl 12431* The standard bounded metric corresponding to generates the same balls as for radii less than . (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
inf

Theorembdmopn 12432* The standard bounded metric corresponding to generates the same topology as . (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
inf

Theoremmopnex 12433* The topology generated by an extended metric can also be generated by a true metric. Thus, "metrizable topologies" can equivalently be defined in terms of metrics or extended metrics. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremmetrest 12434 Two alternate formulations of a subspace topology of a metric space topology. (Contributed by Jeff Hankins, 19-Aug-2009.) (Proof shortened by Mario Carneiro, 5-Jan-2014.)
t

6.2.5  Continuity in metric spaces

Theoremmetcnp3 12435* Two ways to express that is continuous at for metric spaces. Proposition 14-4.2 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

Theoremmetcnp 12436* Two ways to say a mapping from metric to metric is continuous at point . (Contributed by NM, 11-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

Theoremmetcnp2 12437* Two ways to say a mapping from metric to metric is continuous at point . The distance arguments are swapped compared to metcnp 12436 (and Munkres' metcn 12438) for compatibility with df-lm 12141. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcn 12438* Two ways to say a mapping from metric to metric is continuous. Theorem 10.1 of [Munkres] p. 127. The second biconditional argument says that for every positive "epsilon" there is a positive "delta" such that a distance less than delta in maps to a distance less than epsilon in . (Contributed by NM, 15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

Theoremmetcnpi 12439* Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 12436. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcnpi2 12440* Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 12437. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcnpi3 12441* Epsilon-delta property of a metric space function continuous at . A variation of metcnpi2 12440 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcnpd 12442* Two ways to say a mapping from metric to metric is continuous at point . (Contributed by Jim Kingdon, 14-Jun-2023.)

6.2.6  Topology on the reals

Theoremqtopbasss 12443* The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Jim Kingdon, 22-May-2023.)
inf

Theoremqtopbas 12444 The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)

Theoremretopbas 12445 A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)

Theoremretop 12446 The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)

Theoremuniretop 12447 The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)

Theoremretopon 12448 The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
TopOn

Theoremretps 12449 The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
TopSet

Theoremiooretopg 12450 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) (Revised by Jim Kingdon, 23-May-2023.)

Theoremcnmetdval 12451 Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremcnmet 12452 The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)

Theoremcnxmet 12453 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)

Theoremcntoptopon 12454 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
TopOn

Theoremcntoptop 12455 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)

Theoremcnbl0 12456 Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremcnblcld 12457* Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremremetdval 12458 Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.)

Theoremremet 12459 The absolute value metric determines a metric space on the reals. (Contributed by NM, 10-Feb-2007.)

Theoremrexmet 12460 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)

Theorembl2ioo 12461 A ball in terms of an open interval of reals. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremioo2bl 12462 An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

Theoremioo2blex 12463 An open interval of reals in terms of a ball. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremblssioo 12464 The balls of the standard real metric space are included in the open real intervals. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremtgioo 12465 The topology generated by open intervals of reals is the same as the open sets of the standard metric space on the reals. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremtgqioo 12466 The topology generated by open intervals of reals with rational endpoints is the same as the open sets of the standard metric space on the reals. In particular, this proves that the standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremresubmet 12467 The subspace topology induced by a subset of the reals. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Aug-2014.)
t

Theoremtgioo2cntop 12468 The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
t

Theoremrerestcntop 12469 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
t t

6.2.7  Topological definitions using the reals

Syntaxccncf 12470 Extend class notation to include the operation which returns a class of continuous complex functions.

Definitiondf-cncf 12471* Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.)

Theoremcncfval 12472* The value of the continuous complex function operation is the set of continuous functions from to . (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)

Theoremelcncf 12473* Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)

Theoremelcncf2 12474* Version of elcncf 12473 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)

Theoremcncfrss 12475 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremcncfrss2 12476 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremcncff 12477 A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)

Theoremcncfi 12478* Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)

Theoremelcncf1di 12479* Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 26-Nov-2007.)

Theoremelcncf1ii 12480* Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 26-Nov-2007.)

Theoremrescncf 12481 A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)

Theoremcncffvrn 12482 Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)

Theoremcncfss 12483 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)

Theoremclimcncf 12484 Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremabscncf 12485 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremrecncf 12486 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremimcncf 12487 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremcjcncf 12488 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremmulc1cncf 12489* Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremdivccncfap 12490* Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Jim Kingdon, 9-Jan-2023.)
#

Theoremcncfco 12491 The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.)

Theoremcncfmet 12492 Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)

Theoremcncfcncntop 12493 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
t        t

Theoremcncfcn1cntop 12494 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) (Revised by Jim Kingdon, 16-Jun-2023.)

Theoremcncfmptc 12495* A constant function is a continuous function on . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)

Theoremcncfmptid 12496* The identity function is a continuous function on . (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)

Theoremcncfmpt1f 12497* Composition of continuous functions. analogue of cnmpt11f 12234. (Contributed by Mario Carneiro, 3-Sep-2014.)

Theoremaddccncf 12498* Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremcdivcncfap 12499* Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 26-May-2023.)
#        #

Theoremnegcncf 12500* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)

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