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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmstri2 18401 Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremxmstri 18402 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.)

Theoremmstri 18403 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 18404 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremmstri3 18405 Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremmsrtri 18406 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremxmspropd 18407 Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremmspropd 18408 Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)

Theoremsetsmsbas 18409 The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
sSet TopSet

Theoremsetsmsds 18410 The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
sSet TopSet

Theoremsetsmstset 18411 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
sSet TopSet               TopSet

Theoremsetsmstopn 18412 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
sSet TopSet

Theoremsetsxms 18413 The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
sSet TopSet

Theoremsetsms 18414 The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
sSet TopSet

Theoremtmsval 18415 For any metric there is an associated metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp       sSet TopSet

Theoremtmslem 18416 Lemma for tmsbas 18417, tmsds 18418, and tmstopn 18419. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp

Theoremtmsbas 18417 The base set of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp

Theoremtmsds 18418 The metric of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp

Theoremtmstopn 18419 The topology of a constructed metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp

Theoremtmsxms 18420 The constructed metric space is an extended metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp

Theoremtmsms 18421 The constructed metric space is a metric space given a metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp

Theoremimasf1obl 18422 The image of a metric space ball. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremimasf1oxms 18423 The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremimasf1oms 18424 The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremprdsbl 18425* A ball in the product metric for finite index set is the Cartesian product of balls in all coordinates. For infinite index set this is no longer true; instead the correct statement is that a *closed ball* is the product of closed balls in each coordinate (where closed ball means a set of the form in blcld 18439) - for a counterexample the point in whose -th coordinate is is in but is not in the -ball of the product (since ).

The last assumption, , is needed only in the case , when the right side evaluates to and the left evaluates to if and if . (Contributed by Mario Carneiro, 28-Aug-2015.)

s

Theoremmopni 18426* 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 18427* 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 18428* 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 18429 The balls of a metric space are open sets. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)

Theoremunimopn 18430 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 18431 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 18432 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 18433 A ball of a metric space is an open set. (Contributed by NM, 12-Sep-2006.)

Theoremblopn 18434 A ball of a metric space is an open set. (Contributed by NM, 9-Mar-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremneibl 18435* 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 18436 A ball around a point is a neighborhood of the point. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremlpbl 18437* Every ball around a limit point of a subset includes a member of (even if ). (Contributed by NM, 9-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)

Theoremblsscls2 18438* A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)

Theoremblcld 18439* A "closed ball" in a metric space is actually closed. (Contributed by Mario Carneiro, 31-Dec-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)

Theoremblcls 18440* The closure of an open ball in a metric space is contained in the corresponding closed ball. (The converse is not, in general, true; for example, with the discrete metric, the closed ball of radius 1 is the whole space, but the open ball of radius 1 is just a point, whose closure is also a point.) (Contributed by Mario Carneiro, 31-Dec-2013.)

Theoremblsscls 18441 If two concentric balls have different radii, the closure of the smaller one is contained in the larger one. (Contributed by Mario Carneiro, 5-Jan-2014.)

Theoremmetss 18442* 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 18443* 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 Jeffrey Hankins, 21-Jun-2009.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremmetequiv2 18444* 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 18445* Lemma for metss2 18446. (Contributed by Mario Carneiro, 14-Sep-2015.)

Theoremmetss2 18446* 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 18447* The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremstdbdmetval 18448* Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremstdbdxmet 18449* 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.)

Theoremstdbdmet 18450* The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremstdbdbl 18451* The standard bounded metric corresponding to generates the same balls as for radii less than . (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremstdbdmopn 18452* The standard bounded metric corresponding to generates the same topology as . (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremmopnex 18453* 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.)

Theoremmethaus 18454 The topology generated by a metric space is Hausdorff. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)

Theoremmet1stc 18455 The topology generated by a metric space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremmet2ndci 18456 A separable metric space (a metric space with a countable dense subset) is second-countable. (Contributed by Mario Carneiro, 13-Apr-2015.)

Theoremmet2ndc 18457* A metric space is second-countable iff it is separable (has a countable dense subset). (Contributed by Mario Carneiro, 13-Apr-2015.)

Theoremmetrest 18458 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

Theoremressxms 18459 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
s

Theoremressms 18460 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
s

Theoremprdsmslem1 18461 Lemma for prdsms 18465. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremprdsxmslem1 18462 Lemma for prdsms 18465. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremprdsxmslem2 18463* Lemma for prdsxms 18464. The topology generated by the supremum metric is the same as the product topology, when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremprdsxms 18464 The indexed product structure is an extended metric space when the index set is finite. (Although the extended metric is still valid when the index set is infinite, it no longer agrees with the product topology, which is not metrizable in any case.) (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremprdsms 18465 The indexed product structure is a metric space when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theorempwsxms 18466 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theorempwsms 18467 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremxpsxms 18468 A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremxpsms 18469 A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremtmsxps 18470 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

Theoremtmsxpsmopn 18471 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

Theoremtmsxpsval 18472 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

Theoremtmsxpsval2 18473 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

11.4.4  Continuity in metric spaces

Theoremmetcnp3 18474* 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 18475* 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 18476* Two ways to say a mapping from metric to metric is continuous at point . The distance arguments are swapped compared to metcnp 18475 (and Munkres' metcn 18477) for compatibility with df-lm 17229. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcn 18477* 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 18478* Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 18475. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcnpi2 18479* Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 18476. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcnpi3 18480* Epsilon-delta property of a metric space function continuous at . A variation of metcnpi2 18479 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremtxmetcnp 18481* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtxmetcn 18482* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

11.4.5  The uniform structure generated by a metric

Theoremmetuval 18483* Value of the uniform structure generated by metric . (Contributed by Thierry Arnoux, 1-Dec-2017.)
metUnif

Theoremmetustel 18484* Define a filter base generated by a metric . (Contributed by Thierry Arnoux, 22-Nov-2017.)

Theoremmetustss 18485* Range of the elements of the filter base generated by the metric . (Contributed by Thierry Arnoux, 28-Nov-2017.)

Theoremmetustrel 18486* Elements of the filter base generated by the metric are relations. (Contributed by Thierry Arnoux, 28-Nov-2017.)

Theoremmetustto 18487* Any two elements of the filter base generated by the metric can be compared, like for RR+ (i.e. it's totally ordered). (Contributed by Thierry Arnoux, 22-Nov-2017.)

Theoremmetustid 18488* The identity diagonal is included in all elements of the filter base generated by the metric . (Contributed by Thierry Arnoux, 22-Nov-2017.)

Theoremmetustsym 18489* Elements of the filter base generated by the metric are symmetric. (Contributed by Thierry Arnoux, 28-Nov-2017.)

Theoremmetustexhalf 18490* For any element of the filter base generated by the metric , the half element (corresponding to half the distance) is also in this base. (Contributed by Thierry Arnoux, 28-Nov-2017.)

Theoremmetustfbas 18491* The filter base generated by a metric . (Contributed by Thierry Arnoux, 26-Nov-2017.)

Theoremmetust 18492* The uniform structure generated by a metric (Contributed by Thierry Arnoux, 26-Nov-2017.)
UnifOn

Theoremcfilucfil 18493* Given a metric and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 19103. (Contributed by Thierry Arnoux, 29-Nov-2017.)
CauFilu

Theoremmetuust 18494 The uniform structure generated by metric is a uniform structure. (Contributed by Thierry Arnoux, 1-Dec-2017.)
metUnif UnifOn

Theoremcfilucfil2 18495* Given a metric and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 19103. (Contributed by Thierry Arnoux, 1-Dec-2017.)
CauFilumetUnif

Theoremblval2 18496 The ball around a point , alternative definition. (Contributed by Thierry Arnoux, 7-Dec-2017.)

Theoremelbl4 18497 Membership in a ball, alternative definition. (Contributed by Thierry Arnoux, 26-Jan-2018.)

Theoremmetuel 18498* Elementhood in the uniform structure generated by a metric (Contributed by Thierry Arnoux, 8-Dec-2017.)
metUnif

Theoremmetuel2 18499* Elementhood in the uniform structure generated by a metric (Contributed by Thierry Arnoux, 24-Jan-2018.)
metUnif

Theoremmetustbl 18500* The "section" image of an entourage at a point always contains a ball (centered on this point). (Contributed by Thierry Arnoux, 8-Dec-2017.)
metUnif

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