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Theorem List for Metamath Proof Explorer - 18501-18600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremimasf1obl 18501 The image of a metric space ball. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremimasf1oxms 18502 The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremimasf1oms 18503 The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremprdsbl 18504* 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 18518) - 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 18505* 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 18506* 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 18507* 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 18508 The balls of a metric space are open sets. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)

Theoremunimopn 18509 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 18510 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 18511 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 18512 A ball of a metric space is an open set. (Contributed by NM, 12-Sep-2006.)

Theoremblopn 18513 A ball of a metric space is an open set. (Contributed by NM, 9-Mar-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Theoremneibl 18514* 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 18515 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 18516* 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 18517* A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)

Theoremblcld 18518* 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 18519* 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 18520 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 18521* 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 18522* 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 18523* 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 18524* Lemma for metss2 18525. (Contributed by Mario Carneiro, 14-Sep-2015.)

Theoremmetss2 18525* 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 18526* The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremstdbdmetval 18527* Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremstdbdxmet 18528* 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 18529* 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 18530* The standard bounded metric corresponding to generates the same balls as for radii less than . (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremstdbdmopn 18531* The standard bounded metric corresponding to generates the same topology as . (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremmopnex 18532* 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 18533 The topology generated by a metric space is Hausdorff. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)

Theoremmet1stc 18534 The topology generated by a metric space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremmet2ndci 18535 A separable metric space (a metric space with a countable dense subset) is second-countable. (Contributed by Mario Carneiro, 13-Apr-2015.)

Theoremmet2ndc 18536* A metric space is second-countable iff it is separable (has a countable dense subset). (Contributed by Mario Carneiro, 13-Apr-2015.)

Theoremmetrest 18537 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 18538 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
s

Theoremressms 18539 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
s

Theoremprdsmslem1 18540 Lemma for prdsms 18544. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremprdsxmslem1 18541 Lemma for prdsms 18544. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremprdsxmslem2 18542* Lemma for prdsxms 18543. 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 18543 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 18544 The indexed product structure is a metric space when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theorempwsxms 18545 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theorempwsms 18546 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremxpsxms 18547 A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremxpsms 18548 A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
s

Theoremtmsxps 18549 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

Theoremtmsxpsmopn 18550 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

Theoremtmsxpsval 18551 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

Theoremtmsxpsval2 18552 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp s toMetSp

11.4.5  Continuity in metric spaces

Theoremmetcnp3 18553* 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 18554* 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 18555* Two ways to say a mapping from metric to metric is continuous at point . The distance arguments are swapped compared to metcnp 18554 (and Munkres' metcn 18556) for compatibility with df-lm 17276. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcn 18556* 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 18557* Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 18554. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcnpi2 18558* Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 18555. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremmetcnpi3 18559* Epsilon-delta property of a metric space function continuous at . A variation of metcnpi2 18558 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Theoremtxmetcnp 18560* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtxmetcn 18561* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

11.4.6  The uniform structure generated by a metric

TheoremmetuvalOLD 18562* Value of the uniform structure generated by metric . (Contributed by Thierry Arnoux, 1-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
metUnifOLD

Theoremmetuval 18563* Value of the uniform structure generated by metric . (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet metUnif

TheoremmetustelOLD 18564* Define a filter base generated by a metric . (Contributed by Thierry Arnoux, 22-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustel 18565* Define a filter base generated by a metric . (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetustssOLD 18566* Range of the elements of the filter base generated by the metric . (Contributed by Thierry Arnoux, 28-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustss 18567* Range of the elements of the filter base generated by the metric . (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetustrelOLD 18568* Elements of the filter base generated by the metric are relations. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustrel 18569* Elements of the filter base generated by the metric are relations. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetusttoOLD 18570* 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.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustto 18571* 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.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetustidOLD 18572* The identity diagonal is included in all elements of the filter base generated by the metric . (Contributed by Thierry Arnoux, 22-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustid 18573* The identity diagonal is included in all elements of the filter base generated by the metric . (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetustsymOLD 18574* Elements of the filter base generated by the metric are symmetric. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustsym 18575* Elements of the filter base generated by the metric are symmetric. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetustexhalfOLD 18576* 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.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustexhalf 18577* 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.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetustfbasOLD 18578* The filter base generated by a metric . (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmetustfbas 18579* The filter base generated by a metric . (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet

TheoremmetustOLD 18580* The uniform structure generated by a metric (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
UnifOn

Theoremmetust 18581* The uniform structure generated by a metric (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet UnifOn

TheoremcfilucfilOLD 18582* 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 19201. (Contributed by Thierry Arnoux, 29-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
CauFilu

Theoremcfilucfil 18583* 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 19201. (Contributed by Thierry Arnoux, 29-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet CauFilu

TheoremmetuustOLD 18584 The uniform structure generated by metric is a uniform structure. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
metUnifOLD UnifOn

Theoremmetuust 18585 The uniform structure generated by metric is a uniform structure. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet metUnif UnifOn

Theoremcfilucfil2OLD 18586* 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 19201. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
CauFilumetUnifOLD

Theoremcfilucfil2 18587* 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 19201. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet CauFilumetUnif

Theoremblval2 18588 The ball around a point , alternative definition. (Contributed by Thierry Arnoux, 7-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theoremelbl4 18589 Membership in a ball, alternative definition. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

TheoremmetuelOLD 18590* Elementhood in the uniform structure generated by a metric (Contributed by Thierry Arnoux, 8-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
metUnifOLD

Theoremmetuel 18591* Elementhood in the uniform structure generated by a metric (Contributed by Thierry Arnoux, 8-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
PsMet metUnif

Theoremmetuel2 18592* Elementhood in the uniform structure generated by a metric (Contributed by Thierry Arnoux, 24-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)
metUnif       PsMet

TheoremmetustblOLD 18593* The "section" image of an entourage at a point always contains a ball (centered on this point). (Contributed by Thierry Arnoux, 8-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
metUnifOLD

Theoremmetustbl 18594* The "section" image of an entourage at a point always contains a ball (centered on this point). (Contributed by Thierry Arnoux, 8-Dec-2017.)
PsMet metUnif

TheoremmetutopOLD 18595 The topology induced by a uniform structure generated by a metric is that metric's open sets. (Contributed by Thierry Arnoux, 6-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
unifTopmetUnifOLD

Theorempsmetutop 18596 The topology induced by a uniform structure generated by a metric is generated by that metric's open balls. (Contributed by Thierry Arnoux, 6-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet unifTopmetUnif

Theoremxmetutop 18597 The topology induced by a uniform structure generated by an extended metric is that metric's open sets. (Contributed by Thierry Arnoux, 11-Mar-2018.)
unifTopmetUnif

TheoremxmsuspOLD 18598 If the uniform set of a metric space is the uniform structure generated by its metric, then it is a uniform space. (Contributed by Thierry Arnoux, 14-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
UnifSt       metUnifOLD UnifSp

Theoremxmsusp 18599 If the uniform set of a metric space is the uniform structure generated by its metric, then it is a uniform space. (Contributed by Thierry Arnoux, 14-Dec-2017.)
UnifSt       metUnif UnifSp

Theoremrestmetu 18600 The uniform structure generated by the restriction of a metric is its trace. (Contributed by Thierry Arnoux, 18-Dec-2017.)
PsMet metUnift metUnif

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