TOPMETR: Metric Spaces as Topological Spaces

:: Metric Spaces as Topological Spaces - Fundamental Concepts
:: by Agata Darmochwa{\l} and Yatsuka Nakamura
::
:: Received November 21, 1991
:: Copyright (c) 1991-2021 Association of Mizar Users

:: Topological spaces

theorem :: TOPMETR:1

for T being TopStruct
for F being Subset-Family of T holds
( F is Cover of T iff the carrier of T c= union F ) by SETFAM_1:def 11;

theorem :: TOPMETR:2

for T being non empty TopSpace
for A being non empty SubSpace of T st T is T_2 holds
A is T_2

proof end;

theorem Th3: :: TOPMETR:3

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 Th4: :: TOPMETR:4

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:5

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 p9 being Point of (T | P)
for Q9 being Subset of (T | P) st Q9 = Q /\ P & p9 = p holds
Q9 is a_neighborhood of p9

proof end;

theorem :: TOPMETR:6

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:7

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 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 Reflexive discerning V124() triangle for SubSpace of M;

existence

proof end;

end;

registration

let M be non empty MetrSpace;

cluster non empty strict Reflexive discerning V124() triangle for SubSpace of M;

existence

proof end;

end;

theorem Th8: :: TOPMETR:8

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 Th9: :: TOPMETR:9

for r being Real
for M being MetrSpace
for A being SubSpace of M
for x being Point of M
for x9 being Point of A st x = x9 holds
Ball (x9,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 by Def2;

end;

definition

let a, b be Real;
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 Th10: :: TOPMETR:10

for a, b being Real 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 :: 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);

end;

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

theorem Th11: :: TOPMETR:11

for p, q being Point of RealSpace
for x, y being Real st x = p & y = q holds
dist (p,q) = |.(x - y).| by METRIC_1:def 12;

:: Metric spaces and topology

theorem :: TOPMETR:12

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

theorem Th13: :: TOPMETR:13

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 Th14: :: TOPMETR:14

for r being Real
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 by PCOMPS_1:29;

theorem Th15: :: TOPMETR:15

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 st
( r > 0 & Ball (p,r) c= P ) ) by PCOMPS_1:def 4;

definition

let M be MetrStruct ;

attr M is compact means :: TOPMETR:def 5
TopSpaceMetr M is compact ;

end;

:: deftheorem defines compact TOPMETR:def 5 :

theorem :: TOPMETR:16

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

proof end;

:: REAL as topological space

definition

func R^1 -> TopSpace equals :: TOPMETR:def 6
TopSpaceMetr RealSpace;

correctness ;

end;

:: deftheorem defines R^1 TOPMETR:def 6 :

registration

cluster R^1 -> non empty strict ;

coherence ;

end;

theorem :: TOPMETR:17

the carrier of R^1 = REAL ;

definition

let a, b be Real;

func Closed-Interval-TSpace (a,b) -> non empty strict SubSpace of R^1 equals :: TOPMETR:def 7
TopSpaceMetr (Closed-Interval-MSpace (a,b));

coherence by Th13;

end;

:: deftheorem defines Closed-Interval-TSpace TOPMETR:def 7 :

theorem Th18: :: TOPMETR:18

for a, b being Real st a <= b holds
the carrier of (Closed-Interval-TSpace (a,b)) = [.a,b.] by Th10;

theorem Th19: :: TOPMETR:19

for a, b being Real 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 Th20: :: TOPMETR:20

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

proof end;

definition

:: original: I[01]
redefine func I[01] -> SubSpace of R^1 ;
coherence by Th20;

end;

Lm2: for a, b, r being Real st r >= 0 & a + r <= b holds
a <= b

proof end;

theorem :: TOPMETR:21

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;

::Moved from JORDAN16:6, AK 20.02.2006

theorem :: TOPMETR:22

for T being non empty TopSpace
for A, B being Subset of T st A c= B holds
T | A is SubSpace of T | B

proof end;

::Moved from JGRAPH_5:6,7, AK 20.02.2006

theorem Th23: :: TOPMETR:23

for a, b, d, e being Real
for B being Subset of (Closed-Interval-TSpace (d,e)) st d <= a & a <= b & b <= e & B = [.a,b.] holds
Closed-Interval-TSpace (a,b) = (Closed-Interval-TSpace (d,e)) | B

proof end;

theorem :: TOPMETR:24

for a, b being Real
for B being Subset of I[01] st 0 <= a & a <= b & b <= 1 & B = [.a,b.] holds
Closed-Interval-TSpace (a,b) = I[01] | B by Th20, Th23;

:: from BORSUK_6, 2007.04.08, A,T.

definition

let T be 1-sorted ;

attr T is real-membered means :Def8: :: TOPMETR:def 8
the carrier of T is real-membered ;

end;

:: deftheorem Def8 defines real-membered TOPMETR:def 8 :

registration

cluster I[01] -> real-membered ;

coherence by BORSUK_1:40;

end;

:: from RCOMP_3, 2007.04.08, A,T.

registration