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Theorem List for Metamath Proof Explorer - 17801-17900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremunirnbl 17801 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.)
 |-  ( D  e.  ( * Met `  X )  ->  U. ran  ( ball `  D )  =  X )
 
Theoremblin 17802 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.)
 |-  ( ( ( D  e.  ( * Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* ) )  ->  ( ( P (
 ball `  D ) R )  i^i  ( P ( ball `  D ) S ) )  =  ( P ( ball `  D ) if ( R  <_  S ,  R ,  S ) ) )
 
Theoremssbl 17803 The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( ( D  e.  ( * Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR* )  /\  R  <_  S )  ->  ( P ( ball `  D ) R )  C_  ( P ( ball `  D ) S ) )
 
Theoremblss 17804* Any point  P in a ball  B can be centered in another ball that is a subset of  B. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( D  e.  ( * Met `  X )  /\  B  e.  ran  ( ball `  D )  /\  P  e.  B ) 
 ->  E. x  e.  RR+  ( P ( ball `  D ) x )  C_  B )
 
Theoremblssex 17805* Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X )  ->  ( E. x  e.  ran  ( ball `  D ) ( P  e.  x  /\  x  C_  A ) 
 <-> 
 E. r  e.  RR+  ( P ( ball `  D ) r )  C_  A ) )
 
Theoremssblex 17806* 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.)
 |-  ( ( ( D  e.  ( * Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  E. x  e.  RR+  ( x  <  R  /\  ( P ( ball `  D ) x )  C_  ( P ( ball `  D ) S ) ) )
 
Theoremblin2 17807* 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.)
 |-  ( ( ( D  e.  ( * Met `  X )  /\  P  e.  ( B  i^i  C ) )  /\  ( B  e.  ran  ( ball `  D )  /\  C  e.  ran  ( ball `  D ) ) )  ->  E. x  e.  RR+  ( P ( ball `  D ) x )  C_  ( B  i^i  C ) )
 
Theoremblbas 17808 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.)
 |-  ( D  e.  ( * Met `  X )  ->  ran  ( ball `  D )  e.  TopBases )
 
Theoremblres 17809 A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.)
 |-  C  =  ( D  |`  ( Y  X.  Y ) )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  ( X  i^i  Y )  /\  R  e.  RR* )  ->  ( P ( ball `  C ) R )  =  ( ( P ( ball `  D ) R )  i^i  Y ) )
 
Theoremxmeterval 17810 Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |- 
 .~  =  ( `' D " RR )   =>    |-  ( D  e.  ( * Met `  X )  ->  ( A  .~  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( A D B )  e.  RR )
 ) )
 
Theoremxmeter 17811 The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |- 
 .~  =  ( `' D " RR )   =>    |-  ( D  e.  ( * Met `  X )  ->  .~  Er  X )
 
Theoremxmetec 17812 The equivalence classes under the finite separation equivalence relation are infinity balls. Thus (by erdisj 6593) infinity balls are either identical or disjoint, quite unlike the usual situation with Euclidean balls which admit many kinds of overlap. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |- 
 .~  =  ( `' D " RR )   =>    |-  (
 ( D  e.  ( * Met `  X )  /\  P  e.  X ) 
 ->  [ P ]  .~  =  ( P ( ball `  D )  +oo )
 )
 
Theoremblssec 17813 A ball centered at  P is contained in the set of points finitely separated from  P. This is just an application of ssbl 17803 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |- 
 .~  =  ( `' D " RR )   =>    |-  (
 ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  S  e.  RR* )  ->  ( P ( ball `  D ) S )  C_  [ P ]  .~  )
 
Theoremblpnfctr 17814 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.)
 |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D )  +oo ) )  ->  ( P ( ball `  D )  +oo )  =  ( A ( ball `  D )  +oo ) )
 
Theoremxmetresbl 17815 An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 17812, 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  +oo from each other. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  B  =  ( P ( ball `  D ) R )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) )  e.  ( Met `  B ) )
 
11.3.3  Open sets of a metric space
 
Theoremmopnval 17816 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  ( MetOpen `  D
) is the family of all open sets in the metric space determined by the metric  D. By mopntop 17818, the open sets of a metric space form a topology 
J, whose base set is 
U. J by mopnuni 17819. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  =  ( topGen `  ran  ( ball `  D )
 ) )
 
Theoremmopntopon 17817 The set of open sets of a metric space  X is a topology on  X. 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.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  e.  (TopOn `  X ) )
 
Theoremmopntop 17818 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.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  e.  Top )
 
Theoremmopnuni 17819 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.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
 
Theoremelmopn 17820* The defining property of an open set of a metric space. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  ( A  e.  J  <->  ( A  C_  X  /\  A. x  e.  A  E. y  e.  ran  ( ball `  D ) ( x  e.  y  /\  y  C_  A ) ) ) )
 
Theoremmopnfss 17821 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.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  C_  ~P X )
 
Theoremmopnm 17822 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.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  X  e.  J )
 
Theoremelmopn2 17823* A defining property of an open set of a metric space. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  ( A  e.  J  <->  ( A  C_  X  /\  A. x  e.  A  E. y  e.  RR+  ( x ( ball `  D )
 y )  C_  A ) ) )
 
Theoremmopnss 17824 An open set of a metric space is a subspace of its base set. (Contributed by NM, 3-Sep-2006.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  J )  ->  A  C_  X )
 
Theoremisxms 17825 Express the predicate "
<. X ,  D >. is an extended metric space" with underlying set  X and distance function  D. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( TopOpen `  K )   &    |-  X  =  (
 Base `  K )   &    |-  D  =  ( ( dist `  K )  |`  ( X  X.  X ) )   =>    |-  ( K  e.  *
 MetSp 
 <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D ) ) )
 
Theoremisxms2 17826 Express the predicate "
<. X ,  D >. is an extended metric space" with underlying set  X and distance function  D. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( TopOpen `  K )   &    |-  X  =  (
 Base `  K )   &    |-  D  =  ( ( dist `  K )  |`  ( X  X.  X ) )   =>    |-  ( K  e.  *
 MetSp 
 <->  ( D  e.  ( * Met `  X )  /\  J  =  ( MetOpen `  D ) ) )
 
Theoremisms 17827 Express the predicate "
<. X ,  D >. is a metric space" with underlying set  X and distance function  D. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  J  =  ( TopOpen `  K )   &    |-  X  =  (
 Base `  K )   &    |-  D  =  ( ( dist `  K )  |`  ( X  X.  X ) )   =>    |-  ( K  e.  MetSp  <->  ( K  e.  * MetSp  /\  D  e.  ( Met `  X ) ) )
 
Theoremisms2 17828 Express the predicate "
<. X ,  D >. is a metric space" with underlying set  X and distance function  D. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  J  =  ( TopOpen `  K )   &    |-  X  =  (
 Base `  K )   &    |-  D  =  ( ( dist `  K )  |`  ( X  X.  X ) )   =>    |-  ( K  e.  MetSp  <->  ( D  e.  ( Met `  X )  /\  J  =  ( MetOpen `  D )
 ) )
 
Theoremxmstopn 17829 The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  J  =  ( TopOpen `  K )   &    |-  X  =  (
 Base `  K )   &    |-  D  =  ( ( dist `  K )  |`  ( X  X.  X ) )   =>    |-  ( K  e.  *
 MetSp  ->  J  =  (
 MetOpen `  D ) )
 
Theoremmstopn 17830 The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  J  =  ( TopOpen `  K )   &    |-  X  =  (
 Base `  K )   &    |-  D  =  ( ( dist `  K )  |`  ( X  X.  X ) )   =>    |-  ( K  e.  MetSp  ->  J  =  ( MetOpen `  D ) )
 
Theoremxmstps 17831 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  * MetSp  ->  M  e.  TopSp )
 
Theoremmsxms 17832 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  MetSp  ->  M  e.  * MetSp )
 
Theoremmstps 17833 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( M  e.  MetSp  ->  M  e.  TopSp )
 
Theoremxmsxmet 17834 The distance function, suitably truncated, is a metric on  X. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  ( ( dist `  M )  |`  ( X  X.  X ) )   =>    |-  ( M  e.  * MetSp  ->  D  e.  ( * Met `  X )
 )
 
Theoremmsmet 17835 The distance function, suitably truncated, is a metric on  X. (Contributed by Mario Carneiro, 12-Nov-2013.)
 |-  X  =  ( Base `  M )   &    |-  D  =  ( ( dist `  M )  |`  ( X  X.  X ) )   =>    |-  ( M  e.  MetSp  ->  D  e.  ( Met `  X ) )
 
Theoremmsf 17836 Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  X  =  ( Base `  M )   &    |-  D  =  ( ( dist `  M )  |`  ( X  X.  X ) )   =>    |-  ( M  e.  MetSp  ->  D : ( X  X.  X ) --> RR )
 
Theoremxmsxmet2 17837 The distance function, suitably truncated, is a metric on  X. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( M  e.  *
 MetSp  ->  ( D  |`  ( X  X.  X ) )  e.  ( * Met `  X ) )
 
Theoremmsmet2 17838 The distance function, suitably truncated, is a metric on  X. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( M  e.  MetSp  ->  ( D  |`  ( X  X.  X ) )  e.  ( Met `  X ) )
 
Theoremmscl 17839 Closure of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  MetSp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  RR )
 
Theoremxmscl 17840 Closure of the distance function of an extended metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  * MetSp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  RR* )
 
Theoremxmsge0 17841 The distance function in an extended metric space is nonnegative. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  * MetSp  /\  A  e.  X  /\  B  e.  X )  ->  0  <_  ( A D B ) )
 
Theoremxmseq0 17842 The distance function in an extended metric space is symmetric. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  * MetSp  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A D B )  =  0  <->  A  =  B ) )
 
Theoremxmssym 17843 The distance function in an extended metric space is symmetric. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  * MetSp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( B D A ) )
 
Theoremxmstri2 17844 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  * MetSp  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X )
 )  ->  ( A D B )  <_  (
 ( C D A ) + e ( C D B ) ) )
 
Theoremmstri2 17845 Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  MetSp  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X ) )  ->  ( A D B )  <_  ( ( C D A )  +  ( C D B ) ) )
 
Theoremxmstri 17846 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.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  * MetSp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D B )  <_  (
 ( A D C ) + e ( C D B ) ) )
 
Theoremmstri 17847 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.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  MetSp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A D B )  <_  ( ( A D C )  +  ( C D B ) ) )
 
Theoremxmstri3 17848 Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  * MetSp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D B )  <_  (
 ( A D C ) + e ( B D C ) ) )
 
Theoremmstri3 17849 Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  MetSp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A D B )  <_  ( ( A D C )  +  ( B D C ) ) )
 
Theoremmsrtri 17850 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  M )   &    |-  D  =  (
 dist `  M )   =>    |-  ( ( M  e.  MetSp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( abs `  ( ( A D C )  -  ( B D C ) ) )  <_  ( A D B ) )
 
Theoremxmspropd 17851 Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  (
 ( dist `  K )  |`  ( B  X.  B ) )  =  (
 ( dist `  L )  |`  ( B  X.  B ) ) )   &    |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L ) )   =>    |-  ( ph  ->  ( K  e.  * MetSp  <->  L  e.  * MetSp ) )
 
Theoremmspropd 17852 Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  (
 ( dist `  K )  |`  ( B  X.  B ) )  =  (
 ( dist `  L )  |`  ( B  X.  B ) ) )   &    |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L ) )   =>    |-  ( ph  ->  ( K  e.  MetSp  <->  L  e.  MetSp ) )
 
Theoremsetsmsbas 17853 The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  X  =  ( Base `  M )
 )   &    |-  ( ph  ->  D  =  ( ( dist `  M )  |`  ( X  X.  X ) ) )   &    |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )   =>    |-  ( ph  ->  X  =  ( Base `  K )
 )
 
Theoremsetsmsds 17854 The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  X  =  ( Base `  M )
 )   &    |-  ( ph  ->  D  =  ( ( dist `  M )  |`  ( X  X.  X ) ) )   &    |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )   =>    |-  ( ph  ->  ( dist `  M )  =  (
 dist `  K ) )
 
Theoremsetsmstset 17855 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  X  =  ( Base `  M )
 )   &    |-  ( ph  ->  D  =  ( ( dist `  M )  |`  ( X  X.  X ) ) )   &    |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )   &    |-  ( ph  ->  M  e.  V )   =>    |-  ( ph  ->  ( MetOpen `  D )  =  (TopSet `  K ) )
 
Theoremsetsmstopn 17856 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  X  =  ( Base `  M )
 )   &    |-  ( ph  ->  D  =  ( ( dist `  M )  |`  ( X  X.  X ) ) )   &    |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )   &    |-  ( ph  ->  M  e.  V )   =>    |-  ( ph  ->  ( MetOpen `  D )  =  (
 TopOpen `  K ) )
 
Theoremsetsxms 17857 The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  X  =  ( Base `  M )
 )   &    |-  ( ph  ->  D  =  ( ( dist `  M )  |`  ( X  X.  X ) ) )   &    |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )   &    |-  ( ph  ->  M  e.  V )   =>    |-  ( ph  ->  ( K  e.  * MetSp  <->  D  e.  ( * Met `  X )
 ) )
 
Theoremsetsms 17858 The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  X  =  ( Base `  M )
 )   &    |-  ( ph  ->  D  =  ( ( dist `  M )  |`  ( X  X.  X ) ) )   &    |-  ( ph  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )   &    |-  ( ph  ->  M  e.  V )   =>    |-  ( ph  ->  ( K  e.  MetSp  <->  D  e.  ( Met `  X ) ) )
 
Theoremtmsval 17859 For any metric there is an associated metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  M  =  { <. (
 Base `  ndx ) ,  X >. ,  <. ( dist ` 
 ndx ) ,  D >. }   &    |-  K  =  (toMetSp `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )
 
Theoremtmslem 17860 Lemma for tmsbas 17861, tmsds 17862, and tmstopn 17863. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  M  =  { <. (
 Base `  ndx ) ,  X >. ,  <. ( dist ` 
 ndx ) ,  D >. }   &    |-  K  =  (toMetSp `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  ( X  =  (
 Base `  K )  /\  D  =  ( dist `  K )  /\  ( MetOpen `  D )  =  (
 TopOpen `  K ) ) )
 
Theoremtmsbas 17861 The base set of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  K  =  (toMetSp `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  X  =  ( Base `  K ) )
 
Theoremtmsds 17862 The metric of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  K  =  (toMetSp `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  D  =  ( dist `  K ) )
 
Theoremtmstopn 17863 The topology of a constructed metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  K  =  (toMetSp `  D )   &    |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  =  ( TopOpen `  K ) )
 
Theoremtmsxms 17864 The constructed metric space is an extended metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  K  =  (toMetSp `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  K  e.  * MetSp )
 
Theoremtmsms 17865 The constructed metric space is a metric space given a metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  K  =  (toMetSp `  D )   =>    |-  ( D  e.  ( Met `  X )  ->  K  e.  MetSp )
 
Theoremimasf1obl 17866 The image of a metric space ball. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  U )   &    |-  ( ph  ->  E  e.  ( * Met `  V )
 )   &    |-  ( ph  ->  P  e.  V )   &    |-  ( ph  ->  S  e.  RR* )   =>    |-  ( ph  ->  (
 ( F `  P ) ( ball `  D ) S )  =  ( F " ( P ( ball `  E ) S ) ) )
 
Theoremimasf1oxms 17867 The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  * MetSp )   =>    |-  ( ph  ->  U  e.  * MetSp )
 
Theoremimasf1oms 17868 The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  MetSp )   =>    |-  ( ph  ->  U  e.  MetSp )
 
Theoremprdsbl 17869* 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 17883) - for a counterexample the point  p in  RR ^ NN whose  n-th coordinate is  1  -  1  /  n is in  X_ n  e.  NN ball ( 0 ,  1 ) but is not in the  1-ball of the product (since  d ( 0 ,  p )  =  1).

The last assumption,  0  <  A, is needed only in the case  I  =  (/), when the right side evaluates to  { (/) } and the left evaluates to  (/) if  A  <_  0 and  {
(/) } if  0  <  A. (Contributed by Mario Carneiro, 28-Aug-2015.)

 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  V  =  ( Base `  R )   &    |-  E  =  ( ( dist `  R )  |`  ( V  X.  V ) )   &    |-  D  =  ( dist `  Y )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  R  e.  Z )   &    |-  ( ( ph  /\  x  e.  I )  ->  E  e.  ( * Met `  V ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  0  <  A )   =>    |-  ( ph  ->  ( P ( ball `  D ) A )  =  X_ x  e.  I  (
 ( P `  x ) ( ball `  E ) A ) )
 
Theoremmopni 17870* 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.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  J  /\  P  e.  A ) 
 ->  E. x  e.  ran  ( ball `  D )
 ( P  e.  x  /\  x  C_  A ) )
 
Theoremmopni2 17871* 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.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  J  /\  P  e.  A ) 
 ->  E. x  e.  RR+  ( P ( ball `  D ) x )  C_  A )
 
Theoremmopni3 17872* 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.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( ( D  e.  ( * Met `  X )  /\  A  e.  J  /\  P  e.  A )  /\  R  e.  RR+ )  ->  E. x  e.  RR+  ( x  <  R  /\  ( P (
 ball `  D ) x )  C_  A )
 )
 
Theoremblssopn 17873 The balls of a metric space are open sets. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  ran  ( ball `  D )  C_  J )
 
Theoremunimopn 17874 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.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  A  C_  J )  ->  U. A  e.  J )
 
Theoremmopnin 17875 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.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  J  /\  B  e.  J ) 
 ->  ( A  i^i  B )  e.  J )
 
Theoremmopn0 17876 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.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  -> 
 (/)  e.  J )
 
Theoremrnblopn 17877 A ball of a metric space is an open set. (Contributed by NM, 12-Sep-2006.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  B  e.  ran  ( ball `  D )
 )  ->  B  e.  J )
 
Theoremblopn 17878 A ball of a metric space is an open set. (Contributed by NM, 9-Mar-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R )  e.  J )
 
Theoremneibl 17879* The neighborhoods around a point  P of a metric space are those subsets containing a ball around  P. Definition of neighborhood in [Kreyszig] p. 19. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X )  ->  ( N  e.  ( ( nei `  J ) `  { P }
 ) 
 <->  ( N  C_  X  /\  E. r  e.  RR+  ( P ( ball `  D ) r )  C_  N ) ) )
 
Theoremblnei 17880 A ball around a point is a neighborhood of the point. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR+ )  ->  ( P ( ball `  D ) R )  e.  ( ( nei `  J ) `  { P } ) )
 
Theoremlpbl 17881* Every ball around a limit point  P of a subset  S includes a member of  S (even if  P  e/  S). (Contributed by NM, 9-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  P  e.  ( ( limPt `  J ) `  S ) ) 
 /\  R  e.  RR+ )  ->  E. x  e.  S  x  e.  ( P ( ball `  D ) R ) )
 
Theoremblsscls2 17882* A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  S  =  {
 z  e.  X  |  ( P D z ) 
 <_  R }   =>    |-  ( ( ( D  e.  ( * Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  T  e.  RR*  /\  R  <  T ) )  ->  S  C_  ( P ( ball `  D ) T ) )
 
Theoremblcld 17883* A "closed ball" in a metric space is actually closed. (Contributed by Mario Carneiro, 31-Dec-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  S  =  {
 z  e.  X  |  ( P D z ) 
 <_  R }   =>    |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  S  e.  ( Clsd `  J ) )
 
Theoremblcls 17884* 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.)
 |-  J  =  ( MetOpen `  D )   &    |-  S  =  {
 z  e.  X  |  ( P D z ) 
 <_  R }   =>    |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( ( cls `  J ) `  ( P (
 ball `  D ) R ) )  C_  S )
 
Theoremblsscls 17885 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.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( ( D  e.  ( * Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR*  /\  R  <  S ) )  ->  (
 ( cls `  J ) `  ( P ( ball `  D ) R ) )  C_  ( P ( ball `  D ) S ) )
 
Theoremmetss 17886* Two ways of saying that metric  D generates a finer topology than metric  C. (Contributed by Mario Carneiro, 12-Nov-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  X ) )  ->  ( J 
 C_  K  <->  A. x  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( x (
 ball `  D ) s )  C_  ( x ( ball `  C )
 r ) ) )
 
Theoremmetequiv 17887* 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.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  X ) )  ->  ( J  =  K  <->  A. x  e.  X  ( A. r  e.  RR+  E. s  e.  RR+  ( x ( ball `  D )
 s )  C_  ( x ( ball `  C ) r )  /\  A. a  e.  RR+  E. b  e.  RR+  ( x (
 ball `  C ) b )  C_  ( x ( ball `  D )
 a ) ) ) )
 
Theoremmetequiv2 17888* 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.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  X ) )  ->  ( A. x  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
 ball `  D ) s ) )  ->  J  =  K ) )
 
Theoremmetss2lem 17889* Lemma for metss2 17890. (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   &    |-  ( ph  ->  C  e.  ( Met `  X ) )   &    |-  ( ph  ->  D  e.  ( Met `  X )
 )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x C y )  <_  ( R  x.  ( x D y ) ) )   =>    |-  ( ( ph  /\  ( x  e.  X  /\  S  e.  RR+ ) ) 
 ->  ( x ( ball `  D ) ( S 
 /  R ) ) 
 C_  ( x (
 ball `  C ) S ) )
 
Theoremmetss2 17890* If the metric  D is "strongly finer" than  C (meaning that there is a positive real constant 
R such that  C ( x ,  y )  <_  R  x.  D (
x ,  y )), then  D 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.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   &    |-  ( ph  ->  C  e.  ( Met `  X ) )   &    |-  ( ph  ->  D  e.  ( Met `  X )
 )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x C y )  <_  ( R  x.  ( x D y ) ) )   =>    |-  ( ph  ->  J  C_  K )
 
Theoremcomet 17891* The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  ( ph  ->  F : ( 0 [,]  +oo ) --> RR* )   &    |-  ( ( ph  /\  x  e.  ( 0 [,]  +oo ) )  ->  ( ( F `  x )  =  0  <->  x  =  0 ) )   &    |-  ( ( ph  /\  ( x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo ) ) )  ->  ( x  <_  y  ->  ( F `  x ) 
 <_  ( F `  y
 ) ) )   &    |-  (
 ( ph  /\  ( x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo ) ) )  ->  ( F `  ( x + e y ) )  <_  ( ( F `  x ) + e ( F `  y ) ) )   =>    |-  ( ph  ->  ( F  o.  D )  e.  ( * Met `  X )
 )
 
Theoremstdbdmetval 17892* Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   =>    |-  ( ( R  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  if ( ( A C B )  <_  R ,  ( A C B ) ,  R ) )
 
Theoremstdbdxmet 17893* 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.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  D  e.  ( * Met `  X ) )
 
Theoremstdbdmet 17894* The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  R  e.  RR+ )  ->  D  e.  ( Met `  X ) )
 
Theoremstdbdbl 17895* The standard bounded metric corresponding to  C generates the same balls as  C for radii less than  R. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   =>    |-  ( ( ( C  e.  ( * Met `  X )  /\  R  e.  RR*  /\  0  <  R )  /\  ( P  e.  X  /\  S  e.  RR*  /\  S  <_  R ) )  ->  ( P ( ball `  D ) S )  =  ( P ( ball `  C ) S ) )
 
Theoremstdbdmopn 17896* The standard bounded metric corresponding to  C generates the same topology as  C. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   &    |-  J  =  ( MetOpen `  C )   =>    |-  (
 ( C  e.  ( * Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  J  =  ( MetOpen `  D )
 )
 
Theoremmopnex 17897* 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.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  E. d  e.  ( Met `  X ) J  =  ( MetOpen `  d
 ) )
 
Theoremmethaus 17898 The topology generated by a metric space is Hausdorff. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  e.  Haus )
 
Theoremmet1stc 17899 The topology generated by a metric space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  e.  1stc )
 
Theoremmet2ndci 17900 A separable metric space (a metric space with a countable dense subset) is second-countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  ( A  C_  X  /\  A  ~<_  om  /\  ( ( cls `  J ) `  A )  =  X ) )  ->  J  e.  2ndc )
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