Type  Label  Description 
Statement 

Theorem  mettri 12301 
Triangle inequality for the distance function of a metric space.
Definition 141.1(d) of [Gleason] p.
223. (Contributed by NM,
27Aug2006.)



Theorem  xmettri3 12302 
Triangle inequality for the distance function of an extended metric.
(Contributed by Mario Carneiro, 20Aug2015.)



Theorem  mettri3 12303 
Triangle inequality for the distance function of a metric space.
(Contributed by NM, 13Mar2007.)



Theorem  xmetrtri 12304 
One half of the reverse triangle inequality for the distance function of
an extended metric. (Contributed by Mario Carneiro, 4Sep2015.)



Theorem  metrtri 12305 
Reverse triangle inequality for the distance function of a metric space.
(Contributed by Mario Carneiro, 5May2014.) (Revised by Jim Kingdon,
21Apr2023.)



Theorem  metn0 12306 
A metric space is nonempty iff its base set is nonempty. (Contributed
by NM, 4Oct2007.) (Revised by Mario Carneiro, 14Aug2015.)



Theorem  xmetres2 12307 
Restriction of an extended metric. (Contributed by Mario Carneiro,
20Aug2015.)



Theorem  metreslem 12308 
Lemma for metres 12311. (Contributed by Mario Carneiro,
24Aug2015.)



Theorem  metres2 12309 
Lemma for metres 12311. (Contributed by FL, 12Oct2006.) (Proof
shortened by Mario Carneiro, 14Aug2015.)



Theorem  xmetres 12310 
A restriction of an extended metric is an extended metric. (Contributed
by Mario Carneiro, 24Aug2015.)



Theorem  metres 12311 
A restriction of a metric is a metric. (Contributed by NM, 26Aug2007.)
(Revised by Mario Carneiro, 14Aug2015.)



Theorem  0met 12312 
The empty metric. (Contributed by NM, 30Aug2006.) (Revised by Mario
Carneiro, 14Aug2015.)



6.2.3 Metric space balls


Theorem  blfvalps 12313* 
The value of the ball function. (Contributed by NM, 30Aug2006.)
(Revised by Mario Carneiro, 11Nov2013.) (Revised by Thierry Arnoux,
11Feb2018.)

PsMet


Theorem  blfval 12314* 
The value of the ball function. (Contributed by NM, 30Aug2006.)
(Revised by Mario Carneiro, 11Nov2013.) (Proof shortened by Thierry
Arnoux, 11Feb2018.)



Theorem  blex 12315 
A ball is a set. (Contributed by Jim Kingdon, 4May2023.)



Theorem  blvalps 12316* 
The ball around a point is the set of all points whose distance
from is less
than the ball's radius . (Contributed by NM,
31Aug2006.) (Revised by Mario Carneiro, 11Nov2013.) (Revised by
Thierry Arnoux, 11Mar2018.)

PsMet


Theorem  blval 12317* 
The ball around a point is the set of all points whose distance
from is less
than the ball's radius . (Contributed by NM,
31Aug2006.) (Revised by Mario Carneiro, 11Nov2013.)



Theorem  elblps 12318 
Membership in a ball. (Contributed by NM, 2Sep2006.) (Revised by
Mario Carneiro, 11Nov2013.) (Revised by Thierry Arnoux,
11Mar2018.)

PsMet


Theorem  elbl 12319 
Membership in a ball. (Contributed by NM, 2Sep2006.) (Revised by
Mario Carneiro, 11Nov2013.)



Theorem  elbl2ps 12320 
Membership in a ball. (Contributed by NM, 9Mar2007.) (Revised by
Thierry Arnoux, 11Mar2018.)

PsMet


Theorem  elbl2 12321 
Membership in a ball. (Contributed by NM, 9Mar2007.)



Theorem  elbl3ps 12322 
Membership in a ball, with reversed distance function arguments.
(Contributed by NM, 10Nov2007.)

PsMet


Theorem  elbl3 12323 
Membership in a ball, with reversed distance function arguments.
(Contributed by NM, 10Nov2007.)



Theorem  blcomps 12324 
Commute the arguments to the ball function. (Contributed by Mario
Carneiro, 22Jan2014.) (Revised by Thierry Arnoux, 11Mar2018.)

PsMet


Theorem  blcom 12325 
Commute the arguments to the ball function. (Contributed by Mario
Carneiro, 22Jan2014.)



Theorem  xblpnfps 12326 
The infinity ball in an extended metric is the set of all points that
are a finite distance from the center. (Contributed by Mario Carneiro,
23Aug2015.) (Revised by Thierry Arnoux, 11Mar2018.)

PsMet


Theorem  xblpnf 12327 
The infinity ball in an extended metric is the set of all points that
are a finite distance from the center. (Contributed by Mario Carneiro,
23Aug2015.)



Theorem  blpnf 12328 
The infinity ball in a standard metric is just the whole space.
(Contributed by Mario Carneiro, 23Aug2015.)



Theorem  bldisj 12329 
Two balls are disjoint if the centertocenter distance is more than the
sum of the radii. (Contributed by Mario Carneiro, 30Dec2013.)



Theorem  blgt0 12330 
A nonempty ball implies that the radius is positive. (Contributed by
NM, 11Mar2007.) (Revised by Mario Carneiro, 23Aug2015.)



Theorem  bl2in 12331 
Two balls are disjoint if they don't overlap. (Contributed by NM,
11Mar2007.) (Revised by Mario Carneiro, 23Aug2015.)



Theorem  xblss2ps 12332 
One ball is contained in another if the centertocenter distance is
less than the difference of the radii. In this version of blss2 12335 for
extended metrics, we have to assume the balls are a finite distance
apart, or else will not even be in the infinity ball around
.
(Contributed by Mario Carneiro, 23Aug2015.) (Revised by
Thierry Arnoux, 11Mar2018.)

PsMet


Theorem  xblss2 12333 
One ball is contained in another if the centertocenter distance is
less than the difference of the radii. In this version of blss2 12335 for
extended metrics, we have to assume the balls are a finite distance
apart, or else will not even be in the infinity ball around
.
(Contributed by Mario Carneiro, 23Aug2015.)



Theorem  blss2ps 12334 
One ball is contained in another if the centertocenter distance is
less than the difference of the radii. (Contributed by Mario Carneiro,
15Jan2014.) (Revised by Mario Carneiro, 23Aug2015.) (Revised by
Thierry Arnoux, 11Mar2018.)

PsMet 

Theorem  blss2 12335 
One ball is contained in another if the centertocenter distance is
less than the difference of the radii. (Contributed by Mario Carneiro,
15Jan2014.) (Revised by Mario Carneiro, 23Aug2015.)



Theorem  blhalf 12336 
A ball of radius is contained in a ball of radius centered
at any point inside the smaller ball. (Contributed by Jeff Madsen,
2Sep2009.) (Proof shortened by Mario Carneiro, 14Jan2014.)



Theorem  blfps 12337 
Mapping of a ball. (Contributed by NM, 7May2007.) (Revised by Mario
Carneiro, 23Aug2015.) (Revised by Thierry Arnoux, 11Mar2018.)

PsMet 

Theorem  blf 12338 
Mapping of a ball. (Contributed by NM, 7May2007.) (Revised by Mario
Carneiro, 23Aug2015.)



Theorem  blrnps 12339* 
Membership in the range of the ball function. Note that
is the
collection of all balls for metric .
(Contributed by NM, 31Aug2006.) (Revised by Mario Carneiro,
12Nov2013.) (Revised by Thierry Arnoux, 11Mar2018.)

PsMet


Theorem  blrn 12340* 
Membership in the range of the ball function. Note that
is the
collection of all balls for metric .
(Contributed by NM, 31Aug2006.) (Revised by Mario Carneiro,
12Nov2013.)



Theorem  xblcntrps 12341 
A ball contains its center. (Contributed by NM, 2Sep2006.) (Revised
by Mario Carneiro, 12Nov2013.) (Revised by Thierry Arnoux,
11Mar2018.)

PsMet


Theorem  xblcntr 12342 
A ball contains its center. (Contributed by NM, 2Sep2006.) (Revised
by Mario Carneiro, 12Nov2013.)



Theorem  blcntrps 12343 
A ball contains its center. (Contributed by NM, 2Sep2006.) (Revised
by Mario Carneiro, 12Nov2013.) (Revised by Thierry Arnoux,
11Mar2018.)

PsMet


Theorem  blcntr 12344 
A ball contains its center. (Contributed by NM, 2Sep2006.) (Revised
by Mario Carneiro, 12Nov2013.)



Theorem  xblm 12345* 
A ball is inhabited iff the radius is positive. (Contributed by Mario
Carneiro, 23Aug2015.)



Theorem  bln0 12346 
A ball is not empty. It is also inhabited, as seen at blcntr 12344.
(Contributed by NM, 6Oct2007.) (Revised by Mario Carneiro,
12Nov2013.)



Theorem  blelrnps 12347 
A ball belongs to the set of balls of a metric space. (Contributed by
NM, 2Sep2006.) (Revised by Mario Carneiro, 12Nov2013.) (Revised by
Thierry Arnoux, 11Mar2018.)

PsMet


Theorem  blelrn 12348 
A ball belongs to the set of balls of a metric space. (Contributed by
NM, 2Sep2006.) (Revised by Mario Carneiro, 12Nov2013.)



Theorem  blssm 12349 
A ball is a subset of the base set of a metric space. (Contributed by
NM, 31Aug2006.) (Revised by Mario Carneiro, 12Nov2013.)



Theorem  unirnblps 12350 
The union of the set of balls of a metric space is its base set.
(Contributed by NM, 12Sep2006.) (Revised by Mario Carneiro,
12Nov2013.) (Revised by Thierry Arnoux, 11Mar2018.)

PsMet 

Theorem  unirnbl 12351 
The union of the set of balls of a metric space is its base set.
(Contributed by NM, 12Sep2006.) (Revised by Mario Carneiro,
12Nov2013.)



Theorem  blininf 12352 
The intersection of two balls with the same center is the smaller of
them. (Contributed by NM, 1Sep2006.) (Revised by Mario Carneiro,
12Nov2013.)

inf


Theorem  ssblps 12353 
The size of a ball increases monotonically with its radius.
(Contributed by NM, 20Sep2007.) (Revised by Mario Carneiro,
24Aug2015.) (Revised by Thierry Arnoux, 11Mar2018.)

PsMet


Theorem  ssbl 12354 
The size of a ball increases monotonically with its radius.
(Contributed by NM, 20Sep2007.) (Revised by Mario Carneiro,
24Aug2015.)



Theorem  blssps 12355* 
Any point in a ball
can be centered in
another ball that is
a subset of .
(Contributed by NM, 31Aug2006.) (Revised by
Mario Carneiro, 24Aug2015.) (Revised by Thierry Arnoux,
11Mar2018.)

PsMet


Theorem  blss 12356* 
Any point in a ball
can be centered in
another ball that is
a subset of .
(Contributed by NM, 31Aug2006.) (Revised by
Mario Carneiro, 24Aug2015.)



Theorem  blssexps 12357* 
Two ways to express the existence of a ball subset. (Contributed by NM,
5May2007.) (Revised by Mario Carneiro, 12Nov2013.) (Revised by
Thierry Arnoux, 11Mar2018.)

PsMet


Theorem  blssex 12358* 
Two ways to express the existence of a ball subset. (Contributed by NM,
5May2007.) (Revised by Mario Carneiro, 12Nov2013.)



Theorem  ssblex 12359* 
A nested ball exists whose radius is less than any desired amount.
(Contributed by NM, 20Sep2007.) (Revised by Mario Carneiro,
12Nov2013.)



Theorem  blin2 12360* 
Given any two balls and a point in their intersection, there is a ball
contained in the intersection with the given center point. (Contributed
by Mario Carneiro, 12Nov2013.)



Theorem  blbas 12361 
The balls of a metric space form a basis for a topology. (Contributed
by NM, 12Sep2006.) (Revised by Mario Carneiro, 15Jan2014.)



Theorem  blres 12362 
A ball in a restricted metric space. (Contributed by Mario Carneiro,
5Jan2014.)



Theorem  xmeterval 12363 
Value of the "finitely separated" relation. (Contributed by Mario
Carneiro, 24Aug2015.)



Theorem  xmeter 12364 
The "finitely separated" relation is an equivalence relation.
(Contributed by Mario Carneiro, 24Aug2015.)



Theorem  xmetec 12365 
The equivalence classes under the finite separation equivalence relation
are infinity balls. (Contributed by Mario Carneiro, 24Aug2015.)



Theorem  blssec 12366 
A ball centered at is
contained in the set of points finitely
separated from . This is just an application of ssbl 12354
to the
infinity ball. (Contributed by Mario Carneiro, 24Aug2015.)



Theorem  blpnfctr 12367 
The infinity ball in an extended metric acts like an ultrametric ball in
that every point in the ball is also its center. (Contributed by Mario
Carneiro, 21Aug2015.)



Theorem  xmetresbl 12368 
An extended metric restricted to any ball (in particular the infinity
ball) is a proper metric. Together with xmetec 12365, this shows that any
extended metric space can be "factored" into the disjoint
union of
proper metric spaces, with points in the same region measured by that
region's metric, and points in different regions being distance
from each other. (Contributed by Mario Carneiro, 23Aug2015.)



6.2.4 Open sets of a metric space


Theorem  mopnrel 12369 
The class of open sets of a metric space is a relation. (Contributed by
Jim Kingdon, 5May2023.)



Theorem  mopnval 12370 
An open set is a subset of a metric space which includes a ball around
each of its points. Definition 1.32 of [Kreyszig] p. 18. The object
is the family of all open sets in the metric space
determined by the metric . By mopntop 12372, the open sets of a
metric space form a topology , whose base set is by
mopnuni 12373. (Contributed by NM, 1Sep2006.) (Revised
by Mario
Carneiro, 12Nov2013.)



Theorem  mopntopon 12371 
The set of open sets of a metric space is a topology on .
Remark in [Kreyszig] p. 19. This
theorem connects the two concepts and
makes available the theorems for topologies for use with metric spaces.
(Contributed by Mario Carneiro, 24Aug2015.)

TopOn 

Theorem  mopntop 12372 
The set of open sets of a metric space is a topology. (Contributed by
NM, 28Aug2006.) (Revised by Mario Carneiro, 12Nov2013.)



Theorem  mopnuni 12373 
The union of all open sets in a metric space is its underlying set.
(Contributed by NM, 4Sep2006.) (Revised by Mario Carneiro,
12Nov2013.)



Theorem  elmopn 12374* 
The defining property of an open set of a metric space. (Contributed by
NM, 1Sep2006.) (Revised by Mario Carneiro, 12Nov2013.)



Theorem  mopnfss 12375 
The family of open sets of a metric space is a collection of subsets of
the base set. (Contributed by NM, 3Sep2006.) (Revised by Mario
Carneiro, 12Nov2013.)



Theorem  mopnm 12376 
The base set of a metric space is open. Part of Theorem T1 of
[Kreyszig] p. 19. (Contributed by NM,
4Sep2006.) (Revised by Mario
Carneiro, 12Nov2013.)



Theorem  elmopn2 12377* 
A defining property of an open set of a metric space. (Contributed by
NM, 5May2007.) (Revised by Mario Carneiro, 12Nov2013.)



Theorem  mopnss 12378 
An open set of a metric space is a subspace of its base set.
(Contributed by NM, 3Sep2006.)



Theorem  isxms 12379 
Express the predicate " is an extended metric space"
with underlying set and distance function . (Contributed by
Mario Carneiro, 2Sep2015.)



Theorem  isxms2 12380 
Express the predicate " is an extended metric space"
with underlying set and distance function . (Contributed by
Mario Carneiro, 2Sep2015.)



Theorem  isms 12381 
Express the predicate " is a metric space" with
underlying set
and distance function . (Contributed by NM,
27Aug2006.) (Revised by Mario Carneiro, 24Aug2015.)



Theorem  isms2 12382 
Express the predicate " is a metric space" with
underlying set
and distance function . (Contributed by NM,
27Aug2006.) (Revised by Mario Carneiro, 24Aug2015.)



Theorem  xmstopn 12383 
The topology component of an extended metric space coincides with the
topology generated by the metric component. (Contributed by Mario
Carneiro, 26Aug2015.)



Theorem  mstopn 12384 
The topology component of a metric space coincides with the topology
generated by the metric component. (Contributed by Mario Carneiro,
26Aug2015.)



Theorem  xmstps 12385 
An extended metric space is a topological space. (Contributed by Mario
Carneiro, 26Aug2015.)



Theorem  msxms 12386 
A metric space is an extended metric space. (Contributed by Mario
Carneiro, 26Aug2015.)



Theorem  mstps 12387 
A metric space is a topological space. (Contributed by Mario Carneiro,
26Aug2015.)



Theorem  xmsxmet 12388 
The distance function, suitably truncated, is an extended metric on
. (Contributed
by Mario Carneiro, 2Sep2015.)



Theorem  msmet 12389 
The distance function, suitably truncated, is a metric on .
(Contributed by Mario Carneiro, 12Nov2013.)



Theorem  msf 12390 
The distance function of a metric space is a function into the real
numbers. (Contributed by NM, 30Aug2006.) (Revised by Mario Carneiro,
12Nov2013.)



Theorem  xmsxmet2 12391 
The distance function, suitably truncated, is an extended metric on
. (Contributed
by Mario Carneiro, 2Oct2015.)



Theorem  msmet2 12392 
The distance function, suitably truncated, is a metric on .
(Contributed by Mario Carneiro, 2Oct2015.)



Theorem  mscl 12393 
Closure of the distance function of a metric space. (Contributed by NM,
30Aug2006.) (Revised by Mario Carneiro, 2Oct2015.)



Theorem  xmscl 12394 
Closure of the distance function of an extended metric space.
(Contributed by Mario Carneiro, 2Oct2015.)



Theorem  xmsge0 12395 
The distance function in an extended metric space is nonnegative.
(Contributed by Mario Carneiro, 4Oct2015.)



Theorem  xmseq0 12396 
The distance between two points in an extended metric space is zero iff
the two points are identical. (Contributed by Mario Carneiro,
2Oct2015.)



Theorem  xmssym 12397 
The distance function in an extended metric space is symmetric.
(Contributed by Mario Carneiro, 2Oct2015.)



Theorem  xmstri2 12398 
Triangle inequality for the distance function of an extended metric.
(Contributed by Mario Carneiro, 2Oct2015.)



Theorem  mstri2 12399 
Triangle inequality for the distance function of a metric space.
(Contributed by Mario Carneiro, 2Oct2015.)



Theorem  xmstri 12400 
Triangle inequality for the distance function of a metric space.
Definition 141.1(d) of [Gleason] p.
223. (Contributed by Mario
Carneiro, 2Oct2015.)

