Type  Label  Description 
Statement 

Theorem  xmeteq0 12601 
The value of an extended metric is zero iff its arguments are equal.
(Contributed by Mario Carneiro, 20Aug2015.)



Theorem  meteq0 12602 
The value of a metric is zero iff its arguments are equal. Property M2
of [Kreyszig] p. 4. (Contributed by
NM, 30Aug2006.)



Theorem  xmettri2 12603 
Triangle inequality for the distance function of an extended metric.
(Contributed by Mario Carneiro, 20Aug2015.)



Theorem  mettri2 12604 
Triangle inequality for the distance function of a metric space.
(Contributed by NM, 30Aug2006.) (Revised by Mario Carneiro,
20Aug2015.)



Theorem  xmet0 12605 
The distance function of a metric space is zero if its arguments are
equal. Definition 141.1(a) of [Gleason] p. 223. (Contributed by Mario
Carneiro, 20Aug2015.)



Theorem  met0 12606 
The distance function of a metric space is zero if its arguments are
equal. Definition 141.1(a) of [Gleason] p. 223. (Contributed by NM,
30Aug2006.)



Theorem  xmetge0 12607 
The distance function of a metric space is nonnegative. (Contributed by
Mario Carneiro, 20Aug2015.)



Theorem  metge0 12608 
The distance function of a metric space is nonnegative. (Contributed by
NM, 27Aug2006.) (Revised by Mario Carneiro, 14Aug2015.)



Theorem  xmetlecl 12609 
Real closure of an extended metric value that is upper bounded by a
real. (Contributed by Mario Carneiro, 20Aug2015.)



Theorem  xmetsym 12610 
The distance function of an extended metric space is symmetric.
(Contributed by Mario Carneiro, 20Aug2015.)



Theorem  xmetpsmet 12611 
An extended metric is a pseudometric. (Contributed by Thierry Arnoux,
7Feb2018.)

PsMet 

Theorem  xmettpos 12612 
The distance function of an extended metric space is symmetric.
(Contributed by Mario Carneiro, 20Aug2015.)

tpos 

Theorem  metsym 12613 
The distance function of a metric space is symmetric. Definition
141.1(c) of [Gleason] p. 223.
(Contributed by NM, 27Aug2006.)
(Revised by Mario Carneiro, 20Aug2015.)



Theorem  xmettri 12614 
Triangle inequality for the distance function of a metric space.
Definition 141.1(d) of [Gleason] p.
223. (Contributed by Mario
Carneiro, 20Aug2015.)



Theorem  mettri 12615 
Triangle inequality for the distance function of a metric space.
Definition 141.1(d) of [Gleason] p.
223. (Contributed by NM,
27Aug2006.)



Theorem  xmettri3 12616 
Triangle inequality for the distance function of an extended metric.
(Contributed by Mario Carneiro, 20Aug2015.)



Theorem  mettri3 12617 
Triangle inequality for the distance function of a metric space.
(Contributed by NM, 13Mar2007.)



Theorem  xmetrtri 12618 
One half of the reverse triangle inequality for the distance function of
an extended metric. (Contributed by Mario Carneiro, 4Sep2015.)



Theorem  metrtri 12619 
Reverse triangle inequality for the distance function of a metric space.
(Contributed by Mario Carneiro, 5May2014.) (Revised by Jim Kingdon,
21Apr2023.)



Theorem  metn0 12620 
A metric space is nonempty iff its base set is nonempty. (Contributed
by NM, 4Oct2007.) (Revised by Mario Carneiro, 14Aug2015.)



Theorem  xmetres2 12621 
Restriction of an extended metric. (Contributed by Mario Carneiro,
20Aug2015.)



Theorem  metreslem 12622 
Lemma for metres 12625. (Contributed by Mario Carneiro,
24Aug2015.)



Theorem  metres2 12623 
Lemma for metres 12625. (Contributed by FL, 12Oct2006.) (Proof
shortened by Mario Carneiro, 14Aug2015.)



Theorem  xmetres 12624 
A restriction of an extended metric is an extended metric. (Contributed
by Mario Carneiro, 24Aug2015.)



Theorem  metres 12625 
A restriction of a metric is a metric. (Contributed by NM, 26Aug2007.)
(Revised by Mario Carneiro, 14Aug2015.)



Theorem  0met 12626 
The empty metric. (Contributed by NM, 30Aug2006.) (Revised by Mario
Carneiro, 14Aug2015.)



7.2.3 Metric space balls


Theorem  blfvalps 12627* 
The value of the ball function. (Contributed by NM, 30Aug2006.)
(Revised by Mario Carneiro, 11Nov2013.) (Revised by Thierry Arnoux,
11Feb2018.)

PsMet


Theorem  blfval 12628* 
The value of the ball function. (Contributed by NM, 30Aug2006.)
(Revised by Mario Carneiro, 11Nov2013.) (Proof shortened by Thierry
Arnoux, 11Feb2018.)



Theorem  blex 12629 
A ball is a set. (Contributed by Jim Kingdon, 4May2023.)



Theorem  blvalps 12630* 
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 12631* 
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 12632 
Membership in a ball. (Contributed by NM, 2Sep2006.) (Revised by
Mario Carneiro, 11Nov2013.) (Revised by Thierry Arnoux,
11Mar2018.)

PsMet


Theorem  elbl 12633 
Membership in a ball. (Contributed by NM, 2Sep2006.) (Revised by
Mario Carneiro, 11Nov2013.)



Theorem  elbl2ps 12634 
Membership in a ball. (Contributed by NM, 9Mar2007.) (Revised by
Thierry Arnoux, 11Mar2018.)

PsMet


Theorem  elbl2 12635 
Membership in a ball. (Contributed by NM, 9Mar2007.)



Theorem  elbl3ps 12636 
Membership in a ball, with reversed distance function arguments.
(Contributed by NM, 10Nov2007.)

PsMet


Theorem  elbl3 12637 
Membership in a ball, with reversed distance function arguments.
(Contributed by NM, 10Nov2007.)



Theorem  blcomps 12638 
Commute the arguments to the ball function. (Contributed by Mario
Carneiro, 22Jan2014.) (Revised by Thierry Arnoux, 11Mar2018.)

PsMet


Theorem  blcom 12639 
Commute the arguments to the ball function. (Contributed by Mario
Carneiro, 22Jan2014.)



Theorem  xblpnfps 12640 
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 12641 
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 12642 
The infinity ball in a standard metric is just the whole space.
(Contributed by Mario Carneiro, 23Aug2015.)



Theorem  bldisj 12643 
Two balls are disjoint if the centertocenter distance is more than the
sum of the radii. (Contributed by Mario Carneiro, 30Dec2013.)



Theorem  blgt0 12644 
A nonempty ball implies that the radius is positive. (Contributed by
NM, 11Mar2007.) (Revised by Mario Carneiro, 23Aug2015.)



Theorem  bl2in 12645 
Two balls are disjoint if they don't overlap. (Contributed by NM,
11Mar2007.) (Revised by Mario Carneiro, 23Aug2015.)



Theorem  xblss2ps 12646 
One ball is contained in another if the centertocenter distance is
less than the difference of the radii. In this version of blss2 12649 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 12647 
One ball is contained in another if the centertocenter distance is
less than the difference of the radii. In this version of blss2 12649 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 12648 
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 12649 
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 12650 
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 12651 
Mapping of a ball. (Contributed by NM, 7May2007.) (Revised by Mario
Carneiro, 23Aug2015.) (Revised by Thierry Arnoux, 11Mar2018.)

PsMet 

Theorem  blf 12652 
Mapping of a ball. (Contributed by NM, 7May2007.) (Revised by Mario
Carneiro, 23Aug2015.)



Theorem  blrnps 12653* 
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 12654* 
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 12655 
A ball contains its center. (Contributed by NM, 2Sep2006.) (Revised
by Mario Carneiro, 12Nov2013.) (Revised by Thierry Arnoux,
11Mar2018.)

PsMet


Theorem  xblcntr 12656 
A ball contains its center. (Contributed by NM, 2Sep2006.) (Revised
by Mario Carneiro, 12Nov2013.)



Theorem  blcntrps 12657 
A ball contains its center. (Contributed by NM, 2Sep2006.) (Revised
by Mario Carneiro, 12Nov2013.) (Revised by Thierry Arnoux,
11Mar2018.)

PsMet


Theorem  blcntr 12658 
A ball contains its center. (Contributed by NM, 2Sep2006.) (Revised
by Mario Carneiro, 12Nov2013.)



Theorem  xblm 12659* 
A ball is inhabited iff the radius is positive. (Contributed by Mario
Carneiro, 23Aug2015.)



Theorem  bln0 12660 
A ball is not empty. It is also inhabited, as seen at blcntr 12658.
(Contributed by NM, 6Oct2007.) (Revised by Mario Carneiro,
12Nov2013.)



Theorem  blelrnps 12661 
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 12662 
A ball belongs to the set of balls of a metric space. (Contributed by
NM, 2Sep2006.) (Revised by Mario Carneiro, 12Nov2013.)



Theorem  blssm 12663 
A ball is a subset of the base set of a metric space. (Contributed by
NM, 31Aug2006.) (Revised by Mario Carneiro, 12Nov2013.)



Theorem  unirnblps 12664 
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 12665 
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 12666 
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 12667 
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 12668 
The size of a ball increases monotonically with its radius.
(Contributed by NM, 20Sep2007.) (Revised by Mario Carneiro,
24Aug2015.)



Theorem  blssps 12669* 
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 12670* 
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 12671* 
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 12672* 
Two ways to express the existence of a ball subset. (Contributed by NM,
5May2007.) (Revised by Mario Carneiro, 12Nov2013.)



Theorem  ssblex 12673* 
A nested ball exists whose radius is less than any desired amount.
(Contributed by NM, 20Sep2007.) (Revised by Mario Carneiro,
12Nov2013.)



Theorem  blin2 12674* 
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 12675 
The balls of a metric space form a basis for a topology. (Contributed
by NM, 12Sep2006.) (Revised by Mario Carneiro, 15Jan2014.)



Theorem  blres 12676 
A ball in a restricted metric space. (Contributed by Mario Carneiro,
5Jan2014.)



Theorem  xmeterval 12677 
Value of the "finitely separated" relation. (Contributed by Mario
Carneiro, 24Aug2015.)



Theorem  xmeter 12678 
The "finitely separated" relation is an equivalence relation.
(Contributed by Mario Carneiro, 24Aug2015.)



Theorem  xmetec 12679 
The equivalence classes under the finite separation equivalence relation
are infinity balls. (Contributed by Mario Carneiro, 24Aug2015.)



Theorem  blssec 12680 
A ball centered at is
contained in the set of points finitely
separated from . This is just an application of ssbl 12668
to the
infinity ball. (Contributed by Mario Carneiro, 24Aug2015.)



Theorem  blpnfctr 12681 
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 12682 
An extended metric restricted to any ball (in particular the infinity
ball) is a proper metric. Together with xmetec 12679, 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.)



7.2.4 Open sets of a metric space


Theorem  mopnrel 12683 
The class of open sets of a metric space is a relation. (Contributed by
Jim Kingdon, 5May2023.)



Theorem  mopnval 12684 
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 12686, the open sets of a
metric space form a topology , whose base set is by
mopnuni 12687. (Contributed by NM, 1Sep2006.) (Revised
by Mario
Carneiro, 12Nov2013.)



Theorem  mopntopon 12685 
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 12686 
The set of open sets of a metric space is a topology. (Contributed by
NM, 28Aug2006.) (Revised by Mario Carneiro, 12Nov2013.)



Theorem  mopnuni 12687 
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 12688* 
The defining property of an open set of a metric space. (Contributed by
NM, 1Sep2006.) (Revised by Mario Carneiro, 12Nov2013.)



Theorem  mopnfss 12689 
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 12690 
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 12691* 
A defining property of an open set of a metric space. (Contributed by
NM, 5May2007.) (Revised by Mario Carneiro, 12Nov2013.)



Theorem  mopnss 12692 
An open set of a metric space is a subspace of its base set.
(Contributed by NM, 3Sep2006.)



Theorem  isxms 12693 
Express the predicate " is an extended metric space"
with underlying set and distance function . (Contributed by
Mario Carneiro, 2Sep2015.)



Theorem  isxms2 12694 
Express the predicate " is an extended metric space"
with underlying set and distance function . (Contributed by
Mario Carneiro, 2Sep2015.)



Theorem  isms 12695 
Express the predicate " is a metric space" with
underlying set
and distance function . (Contributed by NM,
27Aug2006.) (Revised by Mario Carneiro, 24Aug2015.)



Theorem  isms2 12696 
Express the predicate " is a metric space" with
underlying set
and distance function . (Contributed by NM,
27Aug2006.) (Revised by Mario Carneiro, 24Aug2015.)



Theorem  xmstopn 12697 
The topology component of an extended metric space coincides with the
topology generated by the metric component. (Contributed by Mario
Carneiro, 26Aug2015.)



Theorem  mstopn 12698 
The topology component of a metric space coincides with the topology
generated by the metric component. (Contributed by Mario Carneiro,
26Aug2015.)



Theorem  xmstps 12699 
An extended metric space is a topological space. (Contributed by Mario
Carneiro, 26Aug2015.)



Theorem  msxms 12700 
A metric space is an extended metric space. (Contributed by Mario
Carneiro, 26Aug2015.)

