Article TOPREAL2, MML version 4.99.1005
:: TOPREAL2:th 1
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
st b1 <> b2 & b1 in R^2-unit_square & b2 in R^2-unit_square
holds ex b3, b4 being non empty Element of bool the carrier of TOP-REAL 2 st
b3 is_an_arc_of b1,b2 & b4 is_an_arc_of b1,b2 & R^2-unit_square = b3 \/ b4 & b3 /\ b4 = {b1,b2};
:: TOPREAL2:th 2
theorem
R^2-unit_square is compact(TOP-REAL 2);
:: TOPREAL2:th 3
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3, b4 being non empty Element of bool the carrier of TOP-REAL 2
for b5 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b3,the carrier of (TOP-REAL 2) | b4
st b5 is being_homeomorphism((TOP-REAL 2) | b3, (TOP-REAL 2) | b4) &
b3 is_an_arc_of b1,b2
for b6, b7 being Element of the carrier of TOP-REAL 2
st b6 = b5 . b1 & b7 = b5 . b2
holds b4 is_an_arc_of b6,b7;
:: TOPREAL2:attrnot 1 => TOPREAL2:attr 1
definition
let a1 be Element of bool the carrier of TOP-REAL 2;
attr a1 is being_simple_closed_curve means
ex b1 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | R^2-unit_square,the carrier of (TOP-REAL 2) | a1 st
b1 is being_homeomorphism((TOP-REAL 2) | R^2-unit_square, (TOP-REAL 2) | a1);
end;
:: TOPREAL2:dfs 1
definiens
let a1 be Element of bool the carrier of TOP-REAL 2;
To prove
a1 is being_simple_closed_curve
it is sufficient to prove
thus ex b1 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | R^2-unit_square,the carrier of (TOP-REAL 2) | a1 st
b1 is being_homeomorphism((TOP-REAL 2) | R^2-unit_square, (TOP-REAL 2) | a1);
:: TOPREAL2:def 1
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
b1 is being_simple_closed_curve
iff
ex b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | R^2-unit_square,the carrier of (TOP-REAL 2) | b1 st
b2 is being_homeomorphism((TOP-REAL 2) | R^2-unit_square, (TOP-REAL 2) | b1);
:: TOPREAL2:prednot 1 => TOPREAL2:attr 1
notation
let a1 be Element of bool the carrier of TOP-REAL 2;
synonym a1 is_simple_closed_curve for being_simple_closed_curve;
end;
:: TOPREAL2:funcreg 1
registration
cluster R^2-unit_square -> being_simple_closed_curve;
end;
:: TOPREAL2:exreg 1
registration
cluster non empty being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
end;
:: TOPREAL2:modenot 1
definition
mode Simple_closed_curve is being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
end;
:: TOPREAL2:th 4
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
st b1 is being_simple_closed_curve
holds ex b2, b3 being Element of the carrier of TOP-REAL 2 st
b2 <> b3 & b2 in b1 & b3 in b1;
:: TOPREAL2:th 5
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2 holds
b1 is being_simple_closed_curve
iff
(ex b2, b3 being Element of the carrier of TOP-REAL 2 st
b2 <> b3 & b2 in b1 & b3 in b1) &
(for b2, b3 being Element of the carrier of TOP-REAL 2
st b2 <> b3 & b2 in b1 & b3 in b1
holds ex b4, b5 being non empty Element of bool the carrier of TOP-REAL 2 st
b4 is_an_arc_of b2,b3 & b5 is_an_arc_of b2,b3 & b1 = b4 \/ b5 & b4 /\ b5 = {b2,b3});
:: TOPREAL2:th 6
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2 holds
b1 is being_simple_closed_curve
iff
ex b2, b3 being Element of the carrier of TOP-REAL 2 st
ex b4, b5 being non empty Element of bool the carrier of TOP-REAL 2 st
b2 <> b3 & b2 in b1 & b3 in b1 & b4 is_an_arc_of b2,b3 & b5 is_an_arc_of b2,b3 & b1 = b4 \/ b5 & b4 /\ b5 = {b2,b3};
:: TOPREAL2:condreg 1
registration
cluster being_simple_closed_curve -> non empty compact (Element of bool the carrier of TOP-REAL 2);
end;