:: Families of Subsets, Subspaces and Mappings in Topological Spaces
:: by Agata Darmochwa{\l}
::
:: Received June 21, 1989
:: Copyright (c) 1990-2021 Association of Mizar Users

::
:: A FAMILY OF SETS IN TOPOLOGICAL SPACES
::

theorem :: TOPS_2:1

for T being 1-sorted
for F being Subset-Family of T holds F c= bool ([#] T) ;

theorem Th2: :: TOPS_2:2

for T being 1-sorted
for F being Subset-Family of T
for X being set st X c= F holds
X is Subset-Family of T

proof end;

theorem :: TOPS_2:3

for T being non empty 1-sorted
for F being Subset-Family of T st F is Cover of T holds
F <> {}

proof end;

theorem :: TOPS_2:4

for T being 1-sorted
for F, G being Subset-Family of T holds (union F) \ (union G) c= union (F \ G)

proof end;

theorem :: TOPS_2:5

for T being set
for F being Subset-Family of T holds
( F <> {} iff COMPLEMENT F <> {} )

proof end;

theorem Th6: :: TOPS_2:6

for T being set
for F being Subset-Family of T st F <> {} holds
meet (COMPLEMENT F) = (union F) `

proof end;

theorem Th7: :: TOPS_2:7

for T being set
for F being Subset-Family of T st F <> {} holds
union (COMPLEMENT F) = (meet F) `

proof end;

Lm1: for T being 1-sorted
for F being Subset-Family of T st COMPLEMENT F is finite holds
F is finite

proof end;

theorem Th8: :: TOPS_2:8

for T being 1-sorted
for F being Subset-Family of T holds
( COMPLEMENT F is finite iff F is finite )

proof end;

::
:: CLOSED AND OPEN FAMILIES
::

definition

let T be TopStruct ;
let F be Subset-Family of T;

attr F is open means :: TOPS_2:def 1
for P being Subset of T st P in F holds
P is open ;

attr F is closed means :: TOPS_2:def 2
for P being Subset of T st P in F holds
P is closed ;

end;

:: deftheorem defines open TOPS_2:def 1 :
for T being TopStruct
for F being Subset-Family of T holds
( F is open iff for P being Subset of T st P in F holds
P is open );

:: deftheorem defines closed TOPS_2:def 2 :
for T being TopStruct
for F being Subset-Family of T holds
( F is closed iff for P being Subset of T st P in F holds
P is closed );

theorem Th9: :: TOPS_2:9

for T being TopStruct
for F being Subset-Family of T holds
( F is closed iff COMPLEMENT F is open )

proof end;

theorem :: TOPS_2:10

for T being TopStruct
for F being Subset-Family of T holds
( F is open iff COMPLEMENT F is closed )

proof end;

theorem :: TOPS_2:11

for T being TopStruct
for F, G being Subset-Family of T st F c= G & G is open holds
F is open ;

theorem :: TOPS_2:12

for T being TopStruct
for F, G being Subset-Family of T st F c= G & G is closed holds
F is closed ;

theorem :: TOPS_2:13

for T being TopStruct
for F, G being Subset-Family of T st F is open & G is open holds
F \/ G is open

proof end;

theorem :: TOPS_2:14

for T being TopStruct
for F, G being Subset-Family of T st F is open holds
F /\ G is open

proof end;

theorem :: TOPS_2:15

for T being TopStruct
for F, G being Subset-Family of T st F is open holds
F \ G is open

proof end;

theorem :: TOPS_2:16

for T being TopStruct
for F, G being Subset-Family of T st F is closed & G is closed holds
F \/ G is closed

proof end;

theorem :: TOPS_2:17

for T being TopStruct
for F, G being Subset-Family of T st F is closed holds
F /\ G is closed

proof end;

theorem :: TOPS_2:18

for T being TopStruct
for F, G being Subset-Family of T st F is closed holds
F \ G is closed

proof end;

theorem Th19: :: TOPS_2:19

for GX being TopSpace
for W being Subset-Family of GX st W is open holds
union W is open

proof end;

theorem Th20: :: TOPS_2:20

for GX being TopSpace
for W being Subset-Family of GX st W is open & W is finite holds
meet W is open

proof end;

theorem :: TOPS_2:21

for GX being TopSpace
for W being Subset-Family of GX st W is closed & W is finite holds
union W is closed

proof end;

theorem :: TOPS_2:22

for GX being TopSpace
for W being Subset-Family of GX st W is closed holds
meet W is closed

proof end;

::
:: SUBSPACES OF A TOPOLOGICAL SPACE
::

theorem :: TOPS_2:23

for T being TopStruct
for A being SubSpace of T
for F being Subset-Family of A holds F is Subset-Family of T

proof end;

theorem Th24: :: TOPS_2:24

for T being TopStruct
for A being SubSpace of T
for B being Subset of A holds
( B is open iff ex C being Subset of T st
( C is open & C /\ ([#] A) = B ) )

proof end;

theorem Th25: :: TOPS_2:25

for T being TopStruct
for Q being Subset of T
for A being SubSpace of T st Q is open holds
for P being Subset of A st P = Q holds
P is open

proof end;

theorem Th26: :: TOPS_2:26

for T being TopStruct
for Q being Subset of T
for A being SubSpace of T st Q is closed holds
for P being Subset of A st P = Q holds
P is closed

proof end;

theorem :: TOPS_2:27

for T being TopStruct
for F being Subset-Family of T
for A being SubSpace of T st F is open holds
for G being Subset-Family of A st G = F holds
G is open by Th25;

theorem :: TOPS_2:28

for T being TopStruct
for F being Subset-Family of T
for A being SubSpace of T st F is closed holds
for G being Subset-Family of A st G = F holds
G is closed by Th26;

theorem Th29: :: TOPS_2:29

for T being TopStruct
for M, N being Subset of T holds M /\ N is Subset of (T | N)

proof end;

::
:: RESTRICTION OF A FAMILY
::

definition

let T be TopStruct ;
let P be Subset of T;
let F be Subset-Family of T;

func F | P -> Subset-Family of (T | P) means :Def3: :: TOPS_2:def 3
for Q being Subset of (T | P) holds
( Q in it iff ex R being Subset of T st
( R in F & R /\ P = Q ) );

existence
ex b₁ being Subset-Family of (T | P) st
for Q being Subset of (T | P) holds
( Q in b₁ iff ex R being Subset of T st
( R in F & R /\ P = Q ) )

proof end;

uniqueness
for b₁, b₂ being Subset-Family of (T | P) st ( for Q being Subset of (T | P) holds
( Q in b₁ iff ex R being Subset of T st
( R in F & R /\ P = Q ) ) ) & ( for Q being Subset of (T | P) holds
( Q in b₂ iff ex R being Subset of T st
( R in F & R /\ P = Q ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines | TOPS_2:def 3 :
for T being TopStruct
for P being Subset of T
for F being Subset-Family of T
for b₄ being Subset-Family of (T | P) holds
( b₄ = F | P iff for Q being Subset of (T | P) holds
( Q in b₄ iff ex R being Subset of T st
( R in F & R /\ P = Q ) ) );

theorem :: TOPS_2:30

for T being TopStruct
for M being Subset of T
for F, G being Subset-Family of T st F c= G holds
F | M c= G | M

proof end;

theorem Th31: :: TOPS_2:31

for T being TopStruct
for Q, M being Subset of T
for F being Subset-Family of T st Q in F holds
Q /\ M in F | M

proof end;

theorem :: TOPS_2:32

for T being TopStruct
for Q, M being Subset of T
for F being Subset-Family of T st Q c= union F holds
Q /\ M c= union (F | M)

proof end;

theorem :: TOPS_2:33

for T being TopStruct
for M being Subset of T
for F being Subset-Family of T st M c= union F holds
M = union (F | M)

proof end;

theorem Th34: :: TOPS_2:34

for T being TopStruct
for M being Subset of T
for F being Subset-Family of T holds union (F | M) c= union F

proof end;

theorem :: TOPS_2:35

for T being TopStruct
for M being Subset of T
for F being Subset-Family of T st M c= union (F | M) holds
M c= union F

proof end;

theorem :: TOPS_2:36

for T being TopStruct
for M being Subset of T
for F being Subset-Family of T st F is finite holds
F | M is finite

proof end;

theorem :: TOPS_2:37

for T being TopStruct
for M being Subset of T
for F being Subset-Family of T st F is open holds
F | M is open

proof end;

theorem :: TOPS_2:38

for T being TopStruct
for M being Subset of T
for F being Subset-Family of T st F is closed holds
F | M is closed

proof end;

theorem :: TOPS_2:39

for T being TopStruct
for A being SubSpace of T
for F being Subset-Family of A st F is open holds
ex G being Subset-Family of T st
( G is open & ( for AA being Subset of T st AA = [#] A holds
F = G | AA ) )

proof end;

theorem :: TOPS_2:40

for T being TopStruct
for P being Subset of T
for F being Subset-Family of T ex f being Function st
( dom f = F & rng f = F | P & ( for x being set st x in F holds
for Q being Subset of T st Q = x holds
f . x = Q /\ P ) )

proof end;

theorem Th41: :: TOPS_2:41

for X, Y being 1-sorted
for f being Function of X,Y st ( [#] Y = {} implies [#] X = {} ) holds
f " ([#] Y) = [#] X

proof end;

theorem :: TOPS_2:42

for T being 1-sorted
for S being non empty 1-sorted
for f being Function of T,S
for H being Subset-Family of S holds (" f) .: H is Subset-Family of T

proof end;

theorem Th43: :: TOPS_2:43

for X, Y being TopStruct
for f being Function of X,Y st ( [#] Y = {} implies [#] X = {} ) holds
( f is continuous iff for P being Subset of Y st P is open holds
f " P is open )

proof end;

theorem Th44: :: TOPS_2:44

for T, S being TopSpace
for f being Function of T,S holds
( f is continuous iff for P1 being Subset of S holds Cl (f " P1) c= f " (Cl P1) )

proof end;

theorem Th45: :: TOPS_2:45

for T being TopSpace
for S being non empty TopSpace
for f being Function of T,S holds
( f is continuous iff for P being Subset of T holds f .: (Cl P) c= Cl (f .: P) )

proof end;

theorem Th46: :: TOPS_2:46

for T, V being TopStruct
for S being non empty TopStruct
for f being Function of T,S
for g being Function of S,V st f is continuous & g is continuous holds
g * f is continuous

proof end;

theorem :: TOPS_2:47

for T being TopStruct
for S being non empty TopStruct
for f being Function of T,S
for H being Subset-Family of S st f is continuous & H is open holds
for F being Subset-Family of T st F = (" f) .: H holds
F is open

proof end;

theorem :: TOPS_2:48

for T, S being TopStruct
for f being Function of T,S
for H being Subset-Family of S st f is continuous & H is closed holds
for F being Subset-Family of T st F = (" f) .: H holds
F is closed

proof end;

definition

let S, T be set ;
let f be Function of S,T;
assume A1: f is bijective ;

func f /" -> Function of T,S equals :Def4: :: TOPS_2:def 4
f " ;

coherence
f " is Function of T,S

proof end;

end;

:: deftheorem Def4 defines /" TOPS_2:def 4 :
for S, T being set
for f being Function of S,T st f is bijective holds
f /" = f " ;

notation

let S, T be set ;
let f be Function of S,T;
synonym f " for f /" ;

end;

theorem Th49: :: TOPS_2:49

for T being 1-sorted
for S being non empty 1-sorted
for f being Function of T,S st rng f = [#] S & f is one-to-one holds
( dom (f ") = [#] S & rng (f ") = [#] T )

proof end;

theorem Th50: :: TOPS_2:50

for T, S being 1-sorted
for f being Function of T,S st rng f = [#] S & f is one-to-one holds
f " is one-to-one

proof end;

theorem Th51: :: TOPS_2:51

for T being 1-sorted
for S being non empty 1-sorted
for f being Function of T,S st rng f = [#] S & f is one-to-one holds
(f ") " = f

proof end;

theorem :: TOPS_2:52

for T, S being 1-sorted
for f being Function of T,S st rng f = [#] S & f is one-to-one holds
( (f ") * f = id (dom f) & f * (f ") = id (rng f) )

proof end;

theorem Th53: :: TOPS_2:53

for T being 1-sorted
for S, V being non empty 1-sorted
for f being Function of T,S
for g being Function of S,V st rng f = [#] S & f is one-to-one & dom g = [#] S & rng g = [#] V & g is one-to-one holds
(g * f) " = (f ") * (g ")

proof end;

theorem Th54: :: TOPS_2:54

for T, S being 1-sorted
for f being Function of T,S
for P being Subset of T st rng f = [#] S & f is one-to-one holds
f .: P = (f ") " P

proof end;

theorem Th55: :: TOPS_2:55

for T, S being 1-sorted
for f being Function of T,S
for P1 being Subset of S st rng f = [#] S & f is one-to-one holds
f " P1 = (f ") .: P1

proof end;

::
:: HOMEOMORPHISM
::

definition

let S, T be TopStruct ;
let f be Function of S,T;

attr f is being_homeomorphism means :: TOPS_2:def 5
( dom f = [#] S & rng f = [#] T & f is one-to-one & f is continuous & f " is continuous );

end;

:: deftheorem defines being_homeomorphism TOPS_2:def 5 :
for S, T being TopStruct
for f being Function of S,T holds
( f is being_homeomorphism iff ( dom f = [#] S & rng f = [#] T & f is one-to-one & f is continuous & f " is continuous ) );

theorem :: TOPS_2:56

for T being TopStruct
for S being non empty TopStruct
for f being Function of T,S st f is being_homeomorphism holds
f " is being_homeomorphism by Th49, Th50, Th51;

theorem :: TOPS_2:57

for T, S, V being non empty TopStruct
for f being Function of T,S
for g being Function of S,V st f is being_homeomorphism & g is being_homeomorphism holds
g * f is being_homeomorphism

proof end;

theorem :: TOPS_2:58

for T being TopStruct
for S being non empty TopStruct
for f being Function of T,S holds
( f is being_homeomorphism iff ( dom f = [#] T & rng f = [#] S & f is one-to-one & ( for P being Subset of T holds
( P is closed iff f .: P is closed ) ) ) )

proof end;

theorem :: TOPS_2:59

for T being non empty TopSpace
for S being TopSpace
for f being Function of T,S holds
( f is being_homeomorphism iff ( dom f = [#] T & rng f = [#] S & f is one-to-one & ( for P1 being Subset of S holds f " (Cl P1) = Cl (f " P1) ) ) )

proof end;

theorem :: TOPS_2:60

for T being TopSpace
for S being non empty TopSpace
for f being Function of T,S holds
( f is being_homeomorphism iff ( dom f = [#] T & rng f = [#] S & f is one-to-one & ( for P being Subset of T holds f .: (Cl P) = Cl (f .: P) ) ) )

proof end;

theorem Th61: :: TOPS_2:61

for X, Y being non empty TopSpace
for f being Function of X,Y
for A being Subset of X st f is continuous & A is connected holds
f .: A is connected

proof end;

theorem :: TOPS_2:62

for S, T being non empty TopSpace
for f being Function of S,T
for A being Subset of T st f is being_homeomorphism & A is connected holds
f " A is connected

proof end;

theorem :: TOPS_2:63

for GX being non empty TopSpace st ( for x, y being Point of GX ex GY being non empty TopSpace st
( GY is connected & ex f being Function of GY,GX st
( f is continuous & x in rng f & y in rng f ) ) ) holds
GX is connected

proof end;

:: Added by AK, 2009.09.20

theorem Th64: :: TOPS_2:64

for X being TopStruct
for F being Subset-Family of X holds
( F is open iff F c= the topology of X )

proof end;

theorem :: TOPS_2:65

for X being TopStruct
for F being Subset-Family of X holds
( F is closed iff F c= COMPLEMENT the topology of X )

proof end;

registration

let X be TopStruct ;

cluster the topology of X -> open ;

coherence
the topology of X is open by Th64;

cluster open for Element of bool (bool the carrier of X);

existence
ex b₁ being Subset-Family of X st b₁ is open

proof end;

end;