:: TOPMETR semantic presentation

theorem Th1: :: TOPMETR:1

for T being TopStruct
for F being Subset-Family of T holds
( F is_a_cover_of T iff the carrier of T c= union F )

proof end;

theorem Th2: :: TOPMETR:2

for T being non empty TopSpace
for A being non empty SubSpace of T
for p being Point of A holds p is Point of T

proof end;

theorem :: TOPMETR:3

for T being non empty TopSpace
for A being non empty SubSpace of T st T is_T2 holds
A is_T2

proof end;

theorem Th4: :: TOPMETR:4

for T being TopSpace
for A, B being SubSpace of T st the carrier of A c= the carrier of B holds
A is SubSpace of B

proof end;

theorem :: TOPMETR:5

for T being non empty TopSpace
for P, Q being Subset of T holds
( T | P is SubSpace of T | (P \/ Q) & T | Q is SubSpace of T | (P \/ Q) )

proof end;

theorem :: TOPMETR:6

for T being non empty TopSpace
for p being Point of T
for P being non empty Subset of T st p in P holds
for Q being a_neighborhood of p
for p' being Point of (T | P)
for Q' being Subset of (T | P) st Q' = Q /\ P & p' = p holds
Q' is a_neighborhood of p'

proof end;

theorem :: TOPMETR:7

for A, B, C being TopSpace
for f being Function of A,C st f is continuous & C is SubSpace of B holds
for h being Function of A,B st h = f holds
h is continuous

proof end;

theorem :: TOPMETR:8

for A being TopSpace
for B being non empty TopSpace
for f being Function of A,B
for C being SubSpace of B st f is continuous holds
for h being Function of A,C st h = f holds
h is continuous

proof end;

theorem :: TOPMETR:9

for A, B being TopSpace
for f being Function of A,B
for C being Subset of B st f is continuous holds
for h being Function of A,(B | C) st h = f holds
h is continuous

proof end;

theorem :: TOPMETR:10

for X being TopStruct
for Y being non empty TopStruct
for K0 being Subset of X
for f being Function of X,Y
for g being Function of (X | K0),Y st f is continuous & g = f | K0 holds
g is continuous

proof end;

Lm1: for M being non empty MetrSpace
for p being Point of M
for r being real number st r > 0 holds
p in Ball p,r
by TBSP_1:16;

Lm2: for M being MetrSpace holds MetrStruct(# the carrier of M,the distance of M #) is MetrSpace

proof end;

definition

let M be MetrSpace;
mode SubSpace of M -> MetrSpace means :Def1: :: TOPMETR:def 1
( the carrier of it c= the carrier of M & ( for x, y being Point of it holds the distance of it . x,y = the distance of M . x,y ) );
existence

proof end;

end;

:: deftheorem Def1 defines SubSpace TOPMETR:def 1 :

registration

let M be MetrSpace;
cluster strict SubSpace of M;
existence

proof end;

end;

registration

let M be non empty MetrSpace;
cluster non empty strict SubSpace of M;
existence

proof end;

end;

theorem :: TOPMETR:11

canceled;

theorem Th12: :: TOPMETR:12

for M being non empty MetrSpace
for A being non empty SubSpace of M
for p being Point of A holds p is Point of M

proof end;

theorem Th13: :: TOPMETR:13

for r being real number
for M being MetrSpace
for A being SubSpace of M
for x being Point of M
for x' being Point of A st x = x' holds
Ball x',r = (Ball x,r) /\ the carrier of A

proof end;

definition

let M be non empty MetrSpace;
let A be non empty Subset of M;
func M | A -> strict SubSpace of M means :Def2: :: TOPMETR:def 2
the carrier of it = A;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines | TOPMETR:def 2 :

registration

let M be non empty MetrSpace;
let A be non empty Subset of M;
cluster M | A -> non empty strict ;
coherence

proof end;

end;

definition

let a, b be real number ;
assume A1: a <= b ;
func Closed-Interval-MSpace a,b -> non empty strict SubSpace of RealSpace means :Def3: :: TOPMETR:def 3
for P being non empty Subset of RealSpace st P = [.a,b.] holds
it = RealSpace | P;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def3 defines Closed-Interval-MSpace TOPMETR:def 3 :

theorem Th14: :: TOPMETR:14

for a, b being real number st a <= b holds
the carrier of (Closed-Interval-MSpace a,b) = [.a,b.]

proof end;

definition

let M be MetrStruct ;
let F be Subset-Family of M;
attr F is being_ball-family means :Def4: :: TOPMETR:def 4
for P being set st P in F holds
ex p being Point of M ex r being Real st P = Ball p,r;
pred F is_a_cover_of M means :Def5: :: TOPMETR:def 5
the carrier of M c= union F;

end;

:: deftheorem Def4 defines being_ball-family TOPMETR:def 4 :

:: deftheorem Def5 defines is_a_cover_of TOPMETR:def 5 :

notation

let M be MetrStruct ;
let F be Subset-Family of M;
synonym F is_ball-family for being_ball-family F;

end;

theorem Th15: :: TOPMETR:15

for p, q being Point of RealSpace
for x, y being real number st x = p & y = q holds
dist p,q = abs (x - y)

proof end;

theorem Th16: :: TOPMETR:16

for M being MetrStruct holds
( the carrier of M = the carrier of (TopSpaceMetr M) & the topology of (TopSpaceMetr M) = Family_open_set M )

proof end;

theorem :: TOPMETR:17

canceled;

theorem :: TOPMETR:18

canceled;

theorem Th19: :: TOPMETR:19

for M being non empty MetrSpace
for A being non empty SubSpace of M holds TopSpaceMetr A is SubSpace of TopSpaceMetr M

proof end;

theorem :: TOPMETR:20

for n being Nat
for P being Subset of (TOP-REAL n)
for Q being non empty Subset of (Euclid n) st P = Q holds
(TOP-REAL n) | P = TopSpaceMetr ((Euclid n) | Q)

proof end;

theorem Th21: :: TOPMETR:21

for r being real number
for M being triangle MetrStruct
for p being Point of M
for P being Subset of (TopSpaceMetr M) st P = Ball p,r holds
P is open

proof end;

theorem Th22: :: TOPMETR:22

for M being non empty MetrSpace
for P being Subset of (TopSpaceMetr M) holds
( P is open iff for p being Point of M st p in P holds
ex r being real number st
( r > 0 & Ball p,r c= P ) )

proof end;

definition

let M be MetrStruct ;
attr M is compact means :Def6: :: TOPMETR:def 6
TopSpaceMetr M is compact;

end;

:: deftheorem Def6 defines compact TOPMETR:def 6 :

theorem :: TOPMETR:23

for M being non empty MetrSpace holds
( M is compact iff for F being Subset-Family of M st F is_ball-family & F is_a_cover_of M holds
ex G being Subset-Family of M st
( G c= F & G is_a_cover_of M & G is finite ) )

proof end;

definition

func R^1 -> strict TopSpace equals :: TOPMETR:def 7
TopSpaceMetr RealSpace ;
correctness ;

end;

:: deftheorem defines R^1 TOPMETR:def 7 :

registration

cluster R^1 -> non empty strict ;
coherence ;

end;

theorem Th24: :: TOPMETR:24

the carrier of R^1 = REAL

proof end;

registration

let C be set ;
let f be PartFunc of C,the carrier of R^1 ;
let x be set ;
cluster f . x -> real ;
coherence

proof end;

end;

definition

let a, b be real number ;
func Closed-Interval-TSpace a,b -> non empty strict SubSpace of R^1 equals :: TOPMETR:def 8
TopSpaceMetr (Closed-Interval-MSpace a,b);
coherence by Th19;

end;

:: deftheorem defines Closed-Interval-TSpace TOPMETR:def 8 :

theorem Th25: :: TOPMETR:25

for a, b being real number st a <= b holds
the carrier of (Closed-Interval-TSpace a,b) = [.a,b.]

proof end;

theorem Th26: :: TOPMETR:26

for a, b being real number st a <= b holds
for P being Subset of R^1 st P = [.a,b.] holds
Closed-Interval-TSpace a,b = R^1 | P

proof end;

theorem Th27: :: TOPMETR:27

Closed-Interval-TSpace 0,1 = I[01]

proof end;

definition

:: original: I[01]
redefine func I[01] -> strict SubSpace of R^1 ;
coherence by Th27;

end;

Lm3: for a, b, r being real number st r >= 0 & a + r <= b holds
a <= b

proof end;

theorem :: TOPMETR:28

for f being Function of R^1 ,R^1 st ex a, b being Real st
for x being Real holds f . x = (a * x) + b holds
f is continuous

proof end;