:: TOPGEN_3 semantic presentation
:: deftheorem Def1 defines point-filtered TOPGEN_3:def 1 :
theorem Th1: :: TOPGEN_3:1
theorem Th2: :: TOPGEN_3:2
theorem Th3: :: TOPGEN_3:3
theorem Th4: :: TOPGEN_3:4
Lemma82:
for T1, T2 being TopSpace st ( for A being set holds
( A is open Subset of T1 iff A is open Subset of T2 ) ) holds
( the carrier of T1 = the carrier of T2 & the topology of T1 c= the topology of T2 )
theorem Th5: :: TOPGEN_3:5
Lemma85:
for T1, T2 being TopSpace st ( for A being set holds
( A is closed Subset of T1 iff A is closed Subset of T2 ) ) holds
( the carrier of T1 = the carrier of T2 & the topology of T1 c= the topology of T2 )
theorem Th6: :: TOPGEN_3:6
theorem Th7: :: TOPGEN_3:7
theorem Th8: :: TOPGEN_3:8
theorem Th9: :: TOPGEN_3:9
theorem Th10: :: TOPGEN_3:10
:: deftheorem Def2 defines Sorgenfrey-line TOPGEN_3:def 2 :
Lemma107:
the carrier of Sorgenfrey-line = REAL
by ;
consider BB being Subset-Family of REAL such that
Lemma109:
the topology of Sorgenfrey-line = UniCl BB
and
Lemma110:
BB = { [.x,q.[ where x is Element of REAL , q is Element of REAL : ( x < q & q is rational ) }
by ;
BB c= the topology of Sorgenfrey-line
by , CANTOR_1:1;
then Lemma111:
BB is Basis of Sorgenfrey-line
by , , CANTOR_1:def 2;
theorem Th11: :: TOPGEN_3:11
theorem Th12: :: TOPGEN_3:12
theorem Th13: :: TOPGEN_3:13
theorem Th14: :: TOPGEN_3:14
theorem Th15: :: TOPGEN_3:15
theorem Th16: :: TOPGEN_3:16
theorem Th17: :: TOPGEN_3:17
theorem Th18: :: TOPGEN_3:18
:: deftheorem Def3 defines is_local_minimum_of TOPGEN_3:def 3 :
theorem Th19: :: TOPGEN_3:19
:: deftheorem Def4 defines continuum TOPGEN_3:def 4 :
theorem Th20: :: TOPGEN_3:20
:: deftheorem Def5 defines -powers TOPGEN_3:def 5 :
theorem Th21: :: TOPGEN_3:21
theorem Th22: :: TOPGEN_3:22
theorem Th23: :: TOPGEN_3:23
theorem Th24: :: TOPGEN_3:24
theorem Th25: :: TOPGEN_3:25
theorem Th26: :: TOPGEN_3:26
theorem Th27: :: TOPGEN_3:27
theorem Th28: :: TOPGEN_3:28
theorem Th29: :: TOPGEN_3:29
theorem Th30: :: TOPGEN_3:30
theorem Th31: :: TOPGEN_3:31
theorem Th32: :: TOPGEN_3:32
theorem Th33: :: TOPGEN_3:33
:: deftheorem Def6 defines ClFinTop TOPGEN_3:def 6 :
theorem Th34: :: TOPGEN_3:34
theorem Th35: :: TOPGEN_3:35
theorem Th36: :: TOPGEN_3:36
definition
let X be
set ,
x0 be
set ;
func c2 -PointClTop c1 -> strict TopSpace means :
Def7:
:: TOPGEN_3:def 7
( the
carrier of
it = BB & ( for
A being
Subset of
it holds
Cl A = IFEQ A,
{} ,
A,
(A \/ ({X} /\ BB)) ) );
existence
ex b1 being strict TopSpace st
( the carrier of b1 = X & ( for A being Subset of b1 holds Cl A = IFEQ A,{} ,A,(A \/ ({x0} /\ X)) ) )
correctness
uniqueness
for b1, b2 being strict TopSpace st the carrier of b1 = X & ( for A being Subset of b1 holds Cl A = IFEQ A,{} ,A,(A \/ ({x0} /\ X)) ) & the carrier of b2 = X & ( for A being Subset of b2 holds Cl A = IFEQ A,{} ,A,(A \/ ({x0} /\ X)) ) holds
b1 = b2;
end;
:: deftheorem Def7 defines -PointClTop TOPGEN_3:def 7 :
theorem Th37: :: TOPGEN_3:37
theorem Th38: :: TOPGEN_3:38
theorem Th39: :: TOPGEN_3:39
theorem Th40: :: TOPGEN_3:40
theorem Th41: :: TOPGEN_3:41
definition
let X be
set ,
X0 be
set ;
func c2 -DiscreteTop c1 -> strict TopSpace means :
Def8:
:: TOPGEN_3:def 8
( the
carrier of
it = BB & ( for
A being
Subset of
it holds
Int A = IFEQ A,
BB,
A,
(A /\ X) ) );
existence
ex b1 being strict TopSpace st
( the carrier of b1 = X & ( for A being Subset of b1 holds Int A = IFEQ A,X,A,(A /\ X0) ) )
correctness
uniqueness
for b1, b2 being strict TopSpace st the carrier of b1 = X & ( for A being Subset of b1 holds Int A = IFEQ A,X,A,(A /\ X0) ) & the carrier of b2 = X & ( for A being Subset of b2 holds Int A = IFEQ A,X,A,(A /\ X0) ) holds
b1 = b2;
end;
:: deftheorem Def8 defines -DiscreteTop TOPGEN_3:def 8 :
theorem Th42: :: TOPGEN_3:42
theorem Th43: :: TOPGEN_3:43
Lemma167:
for X being set
for A being Subset of X st A is proper holds
not X is empty
theorem Th44: :: TOPGEN_3:44
theorem Th45: :: TOPGEN_3:45
theorem Th46: :: TOPGEN_3:46