Theorem List for Intuitionistic Logic Explorer - 12301-12400 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | mettri 12301 |
Triangle inequality for the distance function of a metric space.
Definition 14-1.1(d) of [Gleason] p.
223. (Contributed by NM,
27-Aug-2006.)
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Theorem | xmettri3 12302 |
Triangle inequality for the distance function of an extended metric.
(Contributed by Mario Carneiro, 20-Aug-2015.)
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Theorem | mettri3 12303 |
Triangle inequality for the distance function of a metric space.
(Contributed by NM, 13-Mar-2007.)
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Theorem | xmetrtri 12304 |
One half of the reverse triangle inequality for the distance function of
an extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)
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Theorem | metrtri 12305 |
Reverse triangle inequality for the distance function of a metric space.
(Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon,
21-Apr-2023.)
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Theorem | metn0 12306 |
A metric space is nonempty iff its base set is nonempty. (Contributed
by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)
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Theorem | xmetres2 12307 |
Restriction of an extended metric. (Contributed by Mario Carneiro,
20-Aug-2015.)
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Theorem | metreslem 12308 |
Lemma for metres 12311. (Contributed by Mario Carneiro,
24-Aug-2015.)
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Theorem | metres2 12309 |
Lemma for metres 12311. (Contributed by FL, 12-Oct-2006.) (Proof
shortened by Mario Carneiro, 14-Aug-2015.)
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Theorem | xmetres 12310 |
A restriction of an extended metric is an extended metric. (Contributed
by Mario Carneiro, 24-Aug-2015.)
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Theorem | metres 12311 |
A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.)
(Revised by Mario Carneiro, 14-Aug-2015.)
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Theorem | 0met 12312 |
The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario
Carneiro, 14-Aug-2015.)
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6.2.3 Metric space balls
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Theorem | blfvalps 12313* |
The value of the ball function. (Contributed by NM, 30-Aug-2006.)
(Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux,
11-Feb-2018.)
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 PsMet       
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Theorem | blfval 12314* |
The value of the ball function. (Contributed by NM, 30-Aug-2006.)
(Revised by Mario Carneiro, 11-Nov-2013.) (Proof shortened by Thierry
Arnoux, 11-Feb-2018.)
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Theorem | blex 12315 |
A ball is a set. (Contributed by Jim Kingdon, 4-May-2023.)
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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,
31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by
Thierry Arnoux, 11-Mar-2018.)
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  PsMet 
         
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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,
31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
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Theorem | elblps 12318 |
Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by
Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux,
11-Mar-2018.)
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  PsMet 
 
            
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Theorem | elbl 12319 |
Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by
Mario Carneiro, 11-Nov-2013.)
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Theorem | elbl2ps 12320 |
Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by
Thierry Arnoux, 11-Mar-2018.)
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   PsMet     
            
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Theorem | elbl2 12321 |
Membership in a ball. (Contributed by NM, 9-Mar-2007.)
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Theorem | elbl3ps 12322 |
Membership in a ball, with reversed distance function arguments.
(Contributed by NM, 10-Nov-2007.)
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   PsMet     
            
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Theorem | elbl3 12323 |
Membership in a ball, with reversed distance function arguments.
(Contributed by NM, 10-Nov-2007.)
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Theorem | blcomps 12324 |
Commute the arguments to the ball function. (Contributed by Mario
Carneiro, 22-Jan-2014.) (Revised by Thierry Arnoux, 11-Mar-2018.)
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   PsMet     
        
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Theorem | blcom 12325 |
Commute the arguments to the ball function. (Contributed by Mario
Carneiro, 22-Jan-2014.)
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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,
23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
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  PsMet 
             
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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,
23-Aug-2015.)
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Theorem | blpnf 12328 |
The infinity ball in a standard metric is just the whole space.
(Contributed by Mario Carneiro, 23-Aug-2015.)
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Theorem | bldisj 12329 |
Two balls are disjoint if the center-to-center distance is more than the
sum of the radii. (Contributed by Mario Carneiro, 30-Dec-2013.)
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Theorem | blgt0 12330 |
A nonempty ball implies that the radius is positive. (Contributed by
NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
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Theorem | bl2in 12331 |
Two balls are disjoint if they don't overlap. (Contributed by NM,
11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
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Theorem | xblss2ps 12332 |
One ball is contained in another if the center-to-center 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, 23-Aug-2015.) (Revised by
Thierry Arnoux, 11-Mar-2018.)
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 PsMet                     
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Theorem | xblss2 12333 |
One ball is contained in another if the center-to-center 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, 23-Aug-2015.)
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Theorem | blss2ps 12334 |
One ball is contained in another if the center-to-center distance is
less than the difference of the radii. (Contributed by Mario Carneiro,
15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by
Thierry Arnoux, 11-Mar-2018.)
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   PsMet                              |
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Theorem | blss2 12335 |
One ball is contained in another if the center-to-center distance is
less than the difference of the radii. (Contributed by Mario Carneiro,
15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.)
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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,
2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jan-2014.)
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Theorem | blfps 12337 |
Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario
Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
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 PsMet               |
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Theorem | blf 12338 |
Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario
Carneiro, 23-Aug-2015.)
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Theorem | blrnps 12339* |
Membership in the range of the ball function. Note that
    is the
collection of all balls for metric .
(Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro,
12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
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 PsMet  
     
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Theorem | blrn 12340* |
Membership in the range of the ball function. Note that
    is the
collection of all balls for metric .
(Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro,
12-Nov-2013.)
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Theorem | xblcntrps 12341 |
A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised
by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux,
11-Mar-2018.)
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  PsMet 

 
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Theorem | xblcntr 12342 |
A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised
by Mario Carneiro, 12-Nov-2013.)
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Theorem | blcntrps 12343 |
A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised
by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux,
11-Mar-2018.)
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  PsMet 

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Theorem | blcntr 12344 |
A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised
by Mario Carneiro, 12-Nov-2013.)
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Theorem | xblm 12345* |
A ball is inhabited iff the radius is positive. (Contributed by Mario
Carneiro, 23-Aug-2015.)
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Theorem | bln0 12346 |
A ball is not empty. It is also inhabited, as seen at blcntr 12344.
(Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro,
12-Nov-2013.)
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Theorem | blelrnps 12347 |
A ball belongs to the set of balls of a metric space. (Contributed by
NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by
Thierry Arnoux, 11-Mar-2018.)
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  PsMet 
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Theorem | blelrn 12348 |
A ball belongs to the set of balls of a metric space. (Contributed by
NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
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Theorem | blssm 12349 |
A ball is a subset of the base set of a metric space. (Contributed by
NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
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Theorem | unirnblps 12350 |
The union of the set of balls of a metric space is its base set.
(Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro,
12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
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 PsMet         |
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Theorem | unirnbl 12351 |
The union of the set of balls of a metric space is its base set.
(Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro,
12-Nov-2013.)
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Theorem | blininf 12352 |
The intersection of two balls with the same center is the smaller of
them. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro,
12-Nov-2013.)
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                          inf  
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Theorem | ssblps 12353 |
The size of a ball increases monotonically with its radius.
(Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro,
24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
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   PsMet    
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Theorem | ssbl 12354 |
The size of a ball increases monotonically with its radius.
(Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro,
24-Aug-2015.)
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Theorem | blssps 12355* |
Any point in a ball
can be centered in
another ball that is
a subset of .
(Contributed by NM, 31-Aug-2006.) (Revised by
Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux,
11-Mar-2018.)
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  PsMet 
             
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Theorem | blss 12356* |
Any point in a ball
can be centered in
another ball that is
a subset of .
(Contributed by NM, 31-Aug-2006.) (Revised by
Mario Carneiro, 24-Aug-2015.)
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Theorem | blssexps 12357* |
Two ways to express the existence of a ball subset. (Contributed by NM,
5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by
Thierry Arnoux, 11-Mar-2018.)
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  PsMet 
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Theorem | blssex 12358* |
Two ways to express the existence of a ball subset. (Contributed by NM,
5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
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Theorem | ssblex 12359* |
A nested ball exists whose radius is less than any desired amount.
(Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro,
12-Nov-2013.)
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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, 12-Nov-2013.)
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Theorem | blbas 12361 |
The balls of a metric space form a basis for a topology. (Contributed
by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 15-Jan-2014.)
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Theorem | blres 12362 |
A ball in a restricted metric space. (Contributed by Mario Carneiro,
5-Jan-2014.)
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Theorem | xmeterval 12363 |
Value of the "finitely separated" relation. (Contributed by Mario
Carneiro, 24-Aug-2015.)
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Theorem | xmeter 12364 |
The "finitely separated" relation is an equivalence relation.
(Contributed by Mario Carneiro, 24-Aug-2015.)
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Theorem | xmetec 12365 |
The equivalence classes under the finite separation equivalence relation
are infinity balls. (Contributed by Mario Carneiro, 24-Aug-2015.)
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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, 24-Aug-2015.)
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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, 21-Aug-2015.)
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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, 23-Aug-2015.)
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6.2.4 Open sets of a metric space
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Theorem | mopnrel 12369 |
The class of open sets of a metric space is a relation. (Contributed by
Jim Kingdon, 5-May-2023.)
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Theorem | mopnval 12370 |
An open set is a subset of a metric space which includes a ball around
each of its points. Definition 1.3-2 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, 1-Sep-2006.) (Revised
by Mario
Carneiro, 12-Nov-2013.)
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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, 24-Aug-2015.)
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          TopOn    |
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Theorem | mopntop 12372 |
The set of open sets of a metric space is a topology. (Contributed by
NM, 28-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
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Theorem | mopnuni 12373 |
The union of all open sets in a metric space is its underlying set.
(Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro,
12-Nov-2013.)
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Theorem | elmopn 12374* |
The defining property of an open set of a metric space. (Contributed by
NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
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Theorem | mopnfss 12375 |
The family of open sets of a metric space is a collection of subsets of
the base set. (Contributed by NM, 3-Sep-2006.) (Revised by Mario
Carneiro, 12-Nov-2013.)
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Theorem | mopnm 12376 |
The base set of a metric space is open. Part of Theorem T1 of
[Kreyszig] p. 19. (Contributed by NM,
4-Sep-2006.) (Revised by Mario
Carneiro, 12-Nov-2013.)
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Theorem | elmopn2 12377* |
A defining property of an open set of a metric space. (Contributed by
NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
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Theorem | mopnss 12378 |
An open set of a metric space is a subspace of its base set.
(Contributed by NM, 3-Sep-2006.)
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Theorem | isxms 12379 |
Express the predicate "   is an extended metric space"
with underlying set and distance function . (Contributed by
Mario Carneiro, 2-Sep-2015.)
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Theorem | isxms2 12380 |
Express the predicate "   is an extended metric space"
with underlying set and distance function . (Contributed by
Mario Carneiro, 2-Sep-2015.)
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Theorem | isms 12381 |
Express the predicate "   is a metric space" with
underlying set
and distance function . (Contributed by NM,
27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
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Theorem | isms2 12382 |
Express the predicate "   is a metric space" with
underlying set
and distance function . (Contributed by NM,
27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
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Theorem | xmstopn 12383 |
The topology component of an extended metric space coincides with the
topology generated by the metric component. (Contributed by Mario
Carneiro, 26-Aug-2015.)
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Theorem | mstopn 12384 |
The topology component of a metric space coincides with the topology
generated by the metric component. (Contributed by Mario Carneiro,
26-Aug-2015.)
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Theorem | xmstps 12385 |
An extended metric space is a topological space. (Contributed by Mario
Carneiro, 26-Aug-2015.)
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Theorem | msxms 12386 |
A metric space is an extended metric space. (Contributed by Mario
Carneiro, 26-Aug-2015.)
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Theorem | mstps 12387 |
A metric space is a topological space. (Contributed by Mario Carneiro,
26-Aug-2015.)
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Theorem | xmsxmet 12388 |
The distance function, suitably truncated, is an extended metric on
. (Contributed
by Mario Carneiro, 2-Sep-2015.)
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Theorem | msmet 12389 |
The distance function, suitably truncated, is a metric on .
(Contributed by Mario Carneiro, 12-Nov-2013.)
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Theorem | msf 12390 |
The distance function of a metric space is a function into the real
numbers. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro,
12-Nov-2013.)
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Theorem | xmsxmet2 12391 |
The distance function, suitably truncated, is an extended metric on
. (Contributed
by Mario Carneiro, 2-Oct-2015.)
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Theorem | msmet2 12392 |
The distance function, suitably truncated, is a metric on .
(Contributed by Mario Carneiro, 2-Oct-2015.)
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Theorem | mscl 12393 |
Closure of the distance function of a metric space. (Contributed by NM,
30-Aug-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
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Theorem | xmscl 12394 |
Closure of the distance function of an extended metric space.
(Contributed by Mario Carneiro, 2-Oct-2015.)
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Theorem | xmsge0 12395 |
The distance function in an extended metric space is nonnegative.
(Contributed by Mario Carneiro, 4-Oct-2015.)
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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,
2-Oct-2015.)
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Theorem | xmssym 12397 |
The distance function in an extended metric space is symmetric.
(Contributed by Mario Carneiro, 2-Oct-2015.)
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Theorem | xmstri2 12398 |
Triangle inequality for the distance function of an extended metric.
(Contributed by Mario Carneiro, 2-Oct-2015.)
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Theorem | mstri2 12399 |
Triangle inequality for the distance function of a metric space.
(Contributed by Mario Carneiro, 2-Oct-2015.)
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Theorem | xmstri 12400 |
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.)
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