:: Totally Bounded Metric Spaces
:: by Alicia de la Cruz
::
:: Received September 30, 1991
:: Copyright (c) 1991-2011 Association of Mizar Users

begin

theorem Th1: :: TBSP_1:1

for L being Real st 0 < L & L < 1 holds
for n, m being Element of NAT st n <= m holds
L to_power m <= L to_power n

proof end;

theorem Th2: :: TBSP_1:2

for L being Real st 0 < L & L < 1 holds
for k being Element of NAT holds
( L to_power k <= 1 & 0 < L to_power k )

proof end;

theorem Th3: :: TBSP_1:3

for L being Real st 0 < L & L < 1 holds
for s being Real st 0 < s holds
ex n being Element of NAT st L to_power n < s

proof end;

definition

let N be non empty MetrStruct ;
attr N is totally_bounded means :Def1: :: TBSP_1:def 1
for r being Real st r > 0 holds
ex G being Subset-Family of N st
( G is finite & the carrier of N = union G & ( for C being Subset of N st C in G holds
ex w being Element of N st C = Ball (w,r) ) );

end;

:: deftheorem Def1 defines totally_bounded TBSP_1:def 1 :
for N being non empty MetrStruct holds
( N is totally_bounded iff for r being Real st r > 0 holds
ex G being Subset-Family of N st
( G is finite & the carrier of N = union G & ( for C being Subset of N st C in G holds
ex w being Element of N st C = Ball (w,r) ) ) );

Lm1: for N being non empty MetrStruct
for f being Function holds
( f is sequence of N iff ( dom f = NAT & ( for x being set st x in NAT holds
f . x is Element of N ) ) )

proof end;

theorem :: TBSP_1:4

canceled;

theorem :: TBSP_1:5

for N being non empty MetrStruct
for f being Function holds
( f is sequence of N iff ( dom f = NAT & ( for n being Element of NAT holds f . n is Element of N ) ) )

proof end;

definition

let N be non empty MetrStruct ;
let S2 be sequence of N;
canceled;
attr S2 is convergent means :Def3: :: TBSP_1:def 3
ex x being Element of N st
for r being Real st r > 0 holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
dist ((S2 . m),x) < r;

end;

:: deftheorem TBSP_1:def 2 :
canceled;

:: deftheorem Def3 defines convergent TBSP_1:def 3 :
for N being non empty MetrStruct
for S2 being sequence of N holds
( S2 is convergent iff ex x being Element of N st
for r being Real st r > 0 holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
dist ((S2 . m),x) < r );

definition

let M be non empty MetrSpace;
let S1 be sequence of M;
assume A1: S1 is convergent ;
func lim S1 -> Element of M means :: TBSP_1:def 4
for r being Real st r > 0 holds
ex n being Element of NAT st
for m being Element of NAT st m >= n holds
dist ((S1 . m),it) < r;
existence
ex b₁ being Element of M st
for r being Real st r > 0 holds
ex n being Element of NAT st
for m being Element of NAT st m >= n holds
dist ((S1 . m),b₁) < r by A1, Def3;
uniqueness
for b₁, b₂ being Element of M st ( for r being Real st r > 0 holds
ex n being Element of NAT st
for m being Element of NAT st m >= n holds
dist ((S1 . m),b₁) < r ) & ( for r being Real st r > 0 holds
ex n being Element of NAT st
for m being Element of NAT st m >= n holds
dist ((S1 . m),b₂) < r ) holds
b₁ = b₂

proof end;

end;

:: deftheorem defines lim TBSP_1:def 4 :
for M being non empty MetrSpace
for S1 being sequence of M st S1 is convergent holds
for b₃ being Element of M holds
( b₃ = lim S1 iff for r being Real st r > 0 holds
ex n being Element of NAT st
for m being Element of NAT st m >= n holds
dist ((S1 . m),b₃) < r );

definition

let N be non empty MetrStruct ;
let S2 be sequence of N;
attr S2 is Cauchy means :Def5: :: TBSP_1:def 5
for r being Real st r > 0 holds
ex p being Element of NAT st
for n, m being Element of NAT st p <= n & p <= m holds
dist ((S2 . n),(S2 . m)) < r;

end;

:: deftheorem Def5 defines Cauchy TBSP_1:def 5 :
for N being non empty MetrStruct
for S2 being sequence of N holds
( S2 is Cauchy iff for r being Real st r > 0 holds
ex p being Element of NAT st
for n, m being Element of NAT st p <= n & p <= m holds
dist ((S2 . n),(S2 . m)) < r );

definition

let N be non empty MetrStruct ;
attr N is complete means :Def6: :: TBSP_1:def 6
for S2 being sequence of N st S2 is Cauchy holds
S2 is convergent ;

end;

:: deftheorem Def6 defines complete TBSP_1:def 6 :
for N being non empty MetrStruct holds
( N is complete iff for S2 being sequence of N st S2 is Cauchy holds
S2 is convergent );

theorem :: TBSP_1:6

canceled;

definition

let N be non empty MetrStruct ;
let S2 be sequence of N;
let n be Element of NAT ;
:: original: .
redefine func S2 . n -> Element of N;
coherence
S2 . n is Element of N

proof end;

end;

theorem Th7: :: TBSP_1:7

for N being non empty MetrStruct
for S2 being sequence of N st N is triangle & N is symmetric & S2 is convergent holds
S2 is Cauchy

proof end;

registration

let M be non empty symmetric triangle MetrStruct ;
cluster Function-like V30( NAT , the carrier of M) convergent -> Cauchy Element of K10(K11(NAT, the carrier of M));
coherence
for b₁ being sequence of M st b₁ is convergent holds
b₁ is Cauchy by Th7;

end;

theorem Th8: :: TBSP_1:8

for N being non empty symmetric MetrStruct
for S2 being sequence of N holds
( S2 is Cauchy iff for r being Real st r > 0 holds
ex p being Element of NAT st
for n, k being Element of NAT st p <= n holds
dist ((S2 . (n + k)),(S2 . n)) < r )

proof end;

theorem :: TBSP_1:9

for M being non empty MetrSpace
for f being contraction of M st M is complete holds
ex c being Element of M st
( f . c = c & ( for y being Element of M st f . y = y holds
y = c ) )

proof end;

theorem :: TBSP_1:10

for T being non empty Reflexive symmetric triangle MetrStruct st TopSpaceMetr T is compact holds
T is complete

proof end;

theorem :: TBSP_1:11

canceled;

theorem :: TBSP_1:12

for N being non empty MetrStruct st N is Reflexive & N is triangle & TopSpaceMetr N is compact holds
N is totally_bounded

proof end;

definition

let N be non empty MetrStruct ;
canceled;
attr N is bounded means :Def8: :: TBSP_1:def 8
ex r being Real st
( 0 < r & ( for x, y being Point of N holds dist (x,y) <= r ) );
let C be Subset of N;
attr C is bounded means :Def9: :: TBSP_1:def 9
ex r being Real st
( 0 < r & ( for x, y being Point of N st x in C & y in C holds
dist (x,y) <= r ) );

end;

:: deftheorem TBSP_1:def 7 :
canceled;

:: deftheorem Def8 defines bounded TBSP_1:def 8 :
for N being non empty MetrStruct holds
( N is bounded iff ex r being Real st
( 0 < r & ( for x, y being Point of N holds dist (x,y) <= r ) ) );

:: deftheorem Def9 defines bounded TBSP_1:def 9 :
for N being non empty MetrStruct
for C being Subset of N holds
( C is bounded iff ex r being Real st
( 0 < r & ( for x, y being Point of N st x in C & y in C holds
dist (x,y) <= r ) ) );

registration

let A be non empty set ;
cluster DiscreteSpace A -> bounded ;
coherence
DiscreteSpace A is bounded

proof end;

end;

registration

cluster non empty Reflexive discerning symmetric triangle Discerning bounded MetrStruct ;
existence
ex b₁ being non empty MetrSpace st b₁ is bounded

proof end;

end;

theorem :: TBSP_1:13

canceled;

theorem :: TBSP_1:14

canceled;

registration

let N be non empty MetrStruct ;
cluster empty -> bounded Element of K10( the carrier of N);
coherence
for b₁ being Subset of N st b₁ is empty holds
b₁ is bounded

proof end;

end;

registration

let N be non empty MetrStruct ;
cluster bounded Element of K10( the carrier of N);
existence
ex b₁ being Subset of N st b₁ is bounded

proof end;

end;

theorem Th15: :: TBSP_1:15

for N being non empty MetrStruct
for C being Subset of N holds
( ( C <> {} & C is bounded implies ex r being Real ex w being Element of N st
( 0 < r & w in C & ( for z being Point of N st z in C holds
dist (w,z) <= r ) ) ) & ( N is symmetric & N is triangle & ex r being Real ex w being Element of N st
( 0 < r & ( for z being Point of N st z in C holds
dist (w,z) <= r ) ) implies C is bounded ) )

proof end;

theorem Th16: :: TBSP_1:16

for N being non empty MetrStruct
for w being Element of N
for r being real number st N is Reflexive & 0 < r holds
w in Ball (w,r)

proof end;

theorem Th17: :: TBSP_1:17

for T being non empty Reflexive symmetric triangle MetrStruct
for t1 being Element of T
for r being real number st r <= 0 holds
Ball (t1,r) = {}

proof end;

Lm2: for T being non empty Reflexive symmetric triangle MetrStruct
for t1 being Element of T
for r being real number st 0 < r holds
Ball (t1,r) is bounded

proof end;

theorem :: TBSP_1:18

canceled;

theorem :: TBSP_1:19

canceled;

registration

let T be non empty Reflexive symmetric triangle MetrStruct ;
let t1 be Element of T;
let r be real number ;
cluster Ball (t1,r) -> bounded ;
coherence
Ball (t1,r) is bounded

proof end;

end;

theorem Th20: :: TBSP_1:20

for T being non empty Reflexive symmetric triangle MetrStruct
for P, Q being Subset of T st P is bounded & Q is bounded holds
P \/ Q is bounded

proof end;

theorem Th21: :: TBSP_1:21

for N being non empty MetrStruct
for C, D being Subset of N st C is bounded & D c= C holds
D is bounded

proof end;

theorem Th22: :: TBSP_1:22

for T being non empty Reflexive symmetric triangle MetrStruct
for t1 being Element of T
for P being Subset of T st P = {t1} holds
P is bounded

proof end;

theorem Th23: :: TBSP_1:23

for T being non empty Reflexive symmetric triangle MetrStruct
for P being Subset of T st P is finite holds
P is bounded

proof end;

registration

let T be non empty Reflexive symmetric triangle MetrStruct ;
cluster finite -> bounded Element of K10( the carrier of T);
coherence
for b₁ being Subset of T st b₁ is finite holds
b₁ is bounded by Th23;

end;

registration

let T be non empty Reflexive symmetric triangle MetrStruct ;
cluster non empty finite Element of K10( the carrier of T);
existence
ex b₁ being Subset of T st
( b₁ is finite & not b₁ is empty )

proof end;

end;

theorem Th24: :: TBSP_1:24

for T being non empty Reflexive symmetric triangle MetrStruct
for Y being Subset-Family of T st Y is finite & ( for P being Subset of T st P in Y holds
P is bounded ) holds
union Y is bounded

proof end;

theorem Th25: :: TBSP_1:25

for N being non empty MetrStruct holds
( N is bounded iff [#] N is bounded )

proof end;

registration

let N be non empty bounded MetrStruct ;
cluster [#] N -> bounded ;
coherence
[#] N is bounded by Th25;

end;

theorem :: TBSP_1:26

for T being non empty Reflexive symmetric triangle MetrStruct st T is totally_bounded holds
T is bounded

proof end;

definition

let N be non empty Reflexive MetrStruct ;
let C be Subset of N;
assume A1: C is bounded ;
func diameter C -> Real means :Def10: :: TBSP_1:def 10
( ( for x, y being Point of N st x in C & y in C holds
dist (x,y) <= it ) & ( for s being Real st ( for x, y being Point of N st x in C & y in C holds
dist (x,y) <= s ) holds
it <= s ) ) if C <> {}
otherwise it = 0 ;
consistency
for b₁ being Real holds verum ;
existence
( ( C <> {} implies ex b₁ being Real st
( ( for x, y being Point of N st x in C & y in C holds
dist (x,y) <= b₁ ) & ( for s being Real st ( for x, y being Point of N st x in C & y in C holds
dist (x,y) <= s ) holds
b₁ <= s ) ) ) & ( not C <> {} implies ex b₁ being Real st b₁ = 0 ) )

proof end;

uniqueness
for b₁, b₂ being Real holds
( ( C <> {} & ( for x, y being Point of N st x in C & y in C holds
dist (x,y) <= b₁ ) & ( for s being Real st ( for x, y being Point of N st x in C & y in C holds
dist (x,y) <= s ) holds
b₁ <= s ) & ( for x, y being Point of N st x in C & y in C holds
dist (x,y) <= b₂ ) & ( for s being Real st ( for x, y being Point of N st x in C & y in C holds
dist (x,y) <= s ) holds
b₂ <= s ) implies b₁ = b₂ ) & ( not C <> {} & b₁ = 0 & b₂ = 0 implies b₁ = b₂ ) )

proof end;

end;

:: deftheorem Def10 defines diameter TBSP_1:def 10 :
for N being non empty Reflexive MetrStruct
for C being Subset of N st C is bounded holds
for b₃ being Real holds
( ( C <> {} implies ( b₃ = diameter C iff ( ( for x, y being Point of N st x in C & y in C holds
dist (x,y) <= b₃ ) & ( for s being Real st ( for x, y being Point of N st x in C & y in C holds
dist (x,y) <= s ) holds
b₃ <= s ) ) ) ) & ( not C <> {} implies ( b₃ = diameter C iff b₃ = 0 ) ) );

theorem :: TBSP_1:27

canceled;

theorem :: TBSP_1:28

for T being non empty Reflexive symmetric triangle MetrStruct
for x being set
for P being Subset of T st P = {x} holds
diameter P = 0

proof end;

theorem Th29: :: TBSP_1:29

for R being non empty Reflexive MetrStruct
for S being Subset of R st S is bounded holds
0 <= diameter S

proof end;

theorem :: TBSP_1:30

for M being non empty MetrSpace
for A being Subset of M st A <> {} & A is bounded & diameter A = 0 holds
ex g being Point of M st A = {g}

proof end;

theorem :: TBSP_1:31

for T being non empty Reflexive symmetric triangle MetrStruct
for t1 being Element of T
for r being Real st 0 < r holds
diameter (Ball (t1,r)) <= 2 * r

proof end;

theorem :: TBSP_1:32

for R being non empty Reflexive MetrStruct
for S, V being Subset of R st S is bounded & V c= S holds
diameter V <= diameter S

proof end;

theorem :: TBSP_1:33

for T being non empty Reflexive symmetric triangle MetrStruct
for P, Q being Subset of T st P is bounded & Q is bounded & P meets Q holds
diameter (P \/ Q) <= (diameter P) + (diameter Q)

proof end;

theorem :: TBSP_1:34

for T being non empty Reflexive symmetric triangle MetrStruct
for S1 being sequence of T st S1 is Cauchy holds
rng S1 is bounded

proof end;