Article COMPTS_1, MML version 4.99.1005
:: COMPTS_1:prednot 1 => COMPTS_1:pred 1
definition
let a1 be 1-sorted;
let a2 be Element of bool bool the carrier of a1;
let a3 be Element of bool the carrier of a1;
pred A2 is_a_cover_of A3 means
a3 c= union a2;
end;
:: COMPTS_1:dfs 1
definiens
let a1 be 1-sorted;
let a2 be Element of bool bool the carrier of a1;
let a3 be Element of bool the carrier of a1;
To prove
a2 is_a_cover_of a3
it is sufficient to prove
thus a3 c= union a2;
:: COMPTS_1:def 1
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b2 is_a_cover_of b3
iff
b3 c= union b2;
:: COMPTS_1:attrnot 1 => COMPTS_1:attr 1
definition
let a1 be set;
attr a1 is centered means
a1 <> {} &
(for b1 being set
st b1 <> {} & b1 c= a1 & b1 is finite
holds meet b1 <> {});
end;
:: COMPTS_1:dfs 2
definiens
let a1 be set;
To prove
a1 is centered
it is sufficient to prove
thus a1 <> {} &
(for b1 being set
st b1 <> {} & b1 c= a1 & b1 is finite
holds meet b1 <> {});
:: COMPTS_1:def 2
theorem
for b1 being set holds
b1 is centered
iff
b1 <> {} &
(for b2 being set
st b2 <> {} & b2 c= b1 & b2 is finite
holds meet b2 <> {});
:: COMPTS_1:attrnot 2 => COMPTS_1:attr 2
definition
let a1 be TopStruct;
attr a1 is compact means
for b1 being Element of bool bool the carrier of a1
st b1 is_a_cover_of a1 & b1 is open(a1)
holds ex b2 being Element of bool bool the carrier of a1 st
b2 c= b1 & b2 is_a_cover_of a1 & b2 is finite;
end;
:: COMPTS_1:dfs 3
definiens
let a1 be TopStruct;
To prove
a1 is compact
it is sufficient to prove
thus for b1 being Element of bool bool the carrier of a1
st b1 is_a_cover_of a1 & b1 is open(a1)
holds ex b2 being Element of bool bool the carrier of a1 st
b2 c= b1 & b2 is_a_cover_of a1 & b2 is finite;
:: COMPTS_1:def 3
theorem
for b1 being TopStruct holds
b1 is compact
iff
for b2 being Element of bool bool the carrier of b1
st b2 is_a_cover_of b1 & b2 is open(b1)
holds ex b3 being Element of bool bool the carrier of b1 st
b3 c= b2 & b3 is_a_cover_of b1 & b3 is finite;
:: COMPTS_1:attrnot 3 => COMPTS_1:attr 3
definition
let a1 be TopStruct;
attr a1 is being_T2 means
for b1, b2 being Element of the carrier of a1
st b1 <> b2
holds ex b3, b4 being Element of bool the carrier of a1 st
b3 is open(a1) & b4 is open(a1) & b1 in b3 & b2 in b4 & b3 misses b4;
end;
:: COMPTS_1:dfs 4
definiens
let a1 be TopStruct;
To prove
a1 is being_T2
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 <> b2
holds ex b3, b4 being Element of bool the carrier of a1 st
b3 is open(a1) & b4 is open(a1) & b1 in b3 & b2 in b4 & b3 misses b4;
:: COMPTS_1:def 4
theorem
for b1 being TopStruct holds
b1 is being_T2
iff
for b2, b3 being Element of the carrier of b1
st b2 <> b3
holds ex b4, b5 being Element of bool the carrier of b1 st
b4 is open(b1) & b5 is open(b1) & b2 in b4 & b3 in b5 & b4 misses b5;
:: COMPTS_1:attrnot 4 => COMPTS_1:attr 4
definition
let a1 be TopStruct;
attr a1 is being_T3 means
for b1 being Element of the carrier of a1
for b2 being Element of bool the carrier of a1
st b2 <> {} & b2 is closed(a1) & b1 in b2 `
holds ex b3, b4 being Element of bool the carrier of a1 st
b3 is open(a1) & b4 is open(a1) & b1 in b3 & b2 c= b4 & b3 misses b4;
end;
:: COMPTS_1:dfs 5
definiens
let a1 be TopStruct;
To prove
a1 is being_T3
it is sufficient to prove
thus for b1 being Element of the carrier of a1
for b2 being Element of bool the carrier of a1
st b2 <> {} & b2 is closed(a1) & b1 in b2 `
holds ex b3, b4 being Element of bool the carrier of a1 st
b3 is open(a1) & b4 is open(a1) & b1 in b3 & b2 c= b4 & b3 misses b4;
:: COMPTS_1:def 5
theorem
for b1 being TopStruct holds
b1 is being_T3
iff
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 <> {} & b3 is closed(b1) & b2 in b3 `
holds ex b4, b5 being Element of bool the carrier of b1 st
b4 is open(b1) & b5 is open(b1) & b2 in b4 & b3 c= b5 & b4 misses b5;
:: COMPTS_1:attrnot 5 => COMPTS_1:attr 5
definition
let a1 be TopStruct;
attr a1 is being_T4 means
for b1, b2 being Element of bool the carrier of a1
st b1 <> {} & b2 <> {} & b1 is closed(a1) & b2 is closed(a1) & b1 misses b2
holds ex b3, b4 being Element of bool the carrier of a1 st
b3 is open(a1) & b4 is open(a1) & b1 c= b3 & b2 c= b4 & b3 misses b4;
end;
:: COMPTS_1:dfs 6
definiens
let a1 be TopStruct;
To prove
a1 is being_T4
it is sufficient to prove
thus for b1, b2 being Element of bool the carrier of a1
st b1 <> {} & b2 <> {} & b1 is closed(a1) & b2 is closed(a1) & b1 misses b2
holds ex b3, b4 being Element of bool the carrier of a1 st
b3 is open(a1) & b4 is open(a1) & b1 c= b3 & b2 c= b4 & b3 misses b4;
:: COMPTS_1:def 6
theorem
for b1 being TopStruct holds
b1 is being_T4
iff
for b2, b3 being Element of bool the carrier of b1
st b2 <> {} & b3 <> {} & b2 is closed(b1) & b3 is closed(b1) & b2 misses b3
holds ex b4, b5 being Element of bool the carrier of b1 st
b4 is open(b1) & b5 is open(b1) & b2 c= b4 & b3 c= b5 & b4 misses b5;
:: COMPTS_1:attrnot 6 => COMPTS_1:attr 3
notation
let a1 be TopStruct;
synonym Hausdorff for being_T2;
end;
:: COMPTS_1:prednot 2 => COMPTS_1:attr 3
notation
let a1 be TopStruct;
synonym a1 is_T2 for being_T2;
end;
:: COMPTS_1:prednot 3 => COMPTS_1:attr 4
notation
let a1 be TopStruct;
synonym a1 is_T3 for being_T3;
end;
:: COMPTS_1:prednot 4 => COMPTS_1:attr 5
notation
let a1 be TopStruct;
synonym a1 is_T4 for being_T4;
end;
:: COMPTS_1:attrnot 7 => COMPTS_1:attr 6
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is compact means
for b1 being Element of bool bool the carrier of a1
st b1 is_a_cover_of a2 & b1 is open(a1)
holds ex b2 being Element of bool bool the carrier of a1 st
b2 c= b1 & b2 is_a_cover_of a2 & b2 is finite;
end;
:: COMPTS_1:dfs 7
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is compact
it is sufficient to prove
thus for b1 being Element of bool bool the carrier of a1
st b1 is_a_cover_of a2 & b1 is open(a1)
holds ex b2 being Element of bool bool the carrier of a1 st
b2 c= b1 & b2 is_a_cover_of a2 & b2 is finite;
:: COMPTS_1:def 7
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is compact(b1)
iff
for b3 being Element of bool bool the carrier of b1
st b3 is_a_cover_of b2 & b3 is open(b1)
holds ex b4 being Element of bool bool the carrier of b1 st
b4 c= b3 & b4 is_a_cover_of b2 & b4 is finite;
:: COMPTS_1:th 9
theorem
for b1 being TopStruct holds
{} b1 is compact(b1);
:: COMPTS_1:th 10
theorem
for b1 being TopStruct holds
b1 is compact
iff
[#] b1 is compact(b1);
:: COMPTS_1:th 11
theorem
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
st b3 c= [#] b2
holds b3 is compact(b1)
iff
for b4 being Element of bool the carrier of b2
st b4 = b3
holds b4 is compact(b2);
:: COMPTS_1:th 12
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
(b2 = {} implies (b2 is compact(b1)
iff
b1 | b2 is compact)) &
(b1 is TopSpace-like & b2 <> {} implies (b2 is compact(b1)
iff
b1 | b2 is compact));
:: COMPTS_1:th 13
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is compact
iff
for b2 being Element of bool bool the carrier of b1
st b2 is centered & b2 is closed(b1)
holds meet b2 <> {};
:: COMPTS_1:th 14
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is compact
iff
for b2 being Element of bool bool the carrier of b1
st b2 <> {} & b2 is closed(b1) & meet b2 = {}
holds ex b3 being Element of bool bool the carrier of b1 st
b3 <> {} & b3 c= b2 & b3 is finite & meet b3 = {};
:: COMPTS_1:th 15
theorem
for b1 being TopSpace-like TopStruct
st b1 is being_T2
for b2 being Element of bool the carrier of b1
st b2 <> {} & b2 is compact(b1)
for b3 being Element of the carrier of b1
st b3 in b2 `
holds ex b4, b5 being Element of bool the carrier of b1 st
b4 is open(b1) & b5 is open(b1) & b3 in b4 & b2 c= b5 & b4 misses b5;
:: COMPTS_1:th 16
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b1 is being_T2 & b2 is compact(b1)
holds b2 is closed(b1);
:: COMPTS_1:th 17
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b1 is compact & b2 is closed(b1)
holds b2 is compact(b1);
:: COMPTS_1:th 18
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is compact(b1) & b3 c= b2 & b3 is closed(b1)
holds b3 is compact(b1);
:: COMPTS_1:th 19
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is compact(b1) & b3 is compact(b1)
holds b2 \/ b3 is compact(b1);
:: COMPTS_1:th 20
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b1 is being_T2 & b2 is compact(b1) & b3 is compact(b1)
holds b2 /\ b3 is compact(b1);
:: COMPTS_1:th 21
theorem
for b1 being TopSpace-like TopStruct
st b1 is being_T2 & b1 is compact
holds b1 is being_T3;
:: COMPTS_1:th 22
theorem
for b1 being TopSpace-like TopStruct
st b1 is being_T2 & b1 is compact
holds b1 is being_T4;
:: COMPTS_1:th 23
theorem
for b1 being TopStruct
for b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b1 is compact & b3 is continuous(b1, b2) & rng b3 = [#] b2
holds b2 is compact;
:: COMPTS_1:th 24
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being non empty TopStruct
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
st b4 is continuous(b1, b3) & rng b4 = [#] b3 & b2 is compact(b1)
holds b4 .: b2 is compact(b3);
:: COMPTS_1:th 25
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b1 is compact & b2 is being_T2 & rng b3 = [#] b2 & b3 is continuous(b1, b2)
for b4 being Element of bool the carrier of b1
st b4 is closed(b1)
holds b3 .: b4 is closed(b2);
:: COMPTS_1:th 26
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b1 is compact & b2 is being_T2 & dom b3 = [#] b1 & rng b3 = [#] b2 & b3 is one-to-one & b3 is continuous(b1, b2)
holds b3 is being_homeomorphism(b1, b2);
:: COMPTS_1:funcnot 1 => COMPTS_1:func 1
definition
let a1 be set;
func 1TopSp A1 -> TopStruct equals
TopStruct(#a1,[#] bool a1#);
end;
:: COMPTS_1:def 8
theorem
for b1 being set holds
1TopSp b1 = TopStruct(#b1,[#] bool b1#);
:: COMPTS_1:funcreg 1
registration
let a1 be set;
cluster 1TopSp a1 -> strict TopSpace-like;
end;
:: COMPTS_1:funcreg 2
registration
let a1 be non empty set;
cluster 1TopSp a1 -> non empty;
end;
:: COMPTS_1:funcreg 3
registration
let a1 be set;
cluster 1TopSp {a1} -> being_T2;
end;
:: COMPTS_1:exreg 1
registration
cluster non empty TopSpace-like being_T2 TopStruct;
end;
:: COMPTS_1:condreg 1
registration
let a1 be non empty TopSpace-like being_T2 TopStruct;
cluster compact -> closed (Element of bool the carrier of a1);
end;
:: COMPTS_1:exreg 2
registration
let a1 be 1-sorted;
cluster finite Element of bool the carrier of a1;
end;
:: COMPTS_1:th 27
theorem
for b1 being TopSpace-like TopStruct
st the carrier of b1 is finite
holds b1 is compact;
:: COMPTS_1:condreg 2
registration
let a1 be TopSpace-like TopStruct;
cluster finite -> compact (Element of bool the carrier of a1);
end;