Article TEX_3, MML version 4.99.1005
:: TEX_3:th 1
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2,b3 constitute_a_decomposition
holds b2 is not empty
iff
b3 is proper(bool the carrier of b1);
:: TEX_3:th 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2,b3 constitute_a_decomposition
holds b2 is dense(b1)
iff
b3 is boundary(b1);
:: TEX_3:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2,b3 constitute_a_decomposition
holds b2 is boundary(b1)
iff
b3 is dense(b1);
:: TEX_3:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2,b3 constitute_a_decomposition
holds b2 is everywhere_dense(b1)
iff
b3 is nowhere_dense(b1);
:: TEX_3:th 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2,b3 constitute_a_decomposition
holds b2 is nowhere_dense(b1)
iff
b3 is everywhere_dense(b1);
:: TEX_3:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st b2,b3 constitute_a_decomposition
holds b2 is proper(b1) & b3 is proper(b1);
:: TEX_3:th 7
theorem
for b1 being non empty non trivial TopSpace-like TopStruct
for b2 being non empty proper Element of bool the carrier of b1 holds
ex b3 being non empty strict proper SubSpace of b1 st
b2 = the carrier of b3;
:: TEX_3:th 8
theorem
for b1 being non empty non trivial TopSpace-like TopStruct
for b2 being non empty proper SubSpace of b1 holds
ex b3 being non empty strict proper SubSpace of b1 st
b2,b3 constitute_a_decomposition;
:: TEX_3:attrnot 1 => TEX_3:attr 1
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be SubSpace of a1;
attr a2 is dense means
for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
holds b1 is dense(a1);
end;
:: TEX_3:dfs 1
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be SubSpace of a1;
To prove
a2 is dense
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
holds b1 is dense(a1);
:: TEX_3:def 1
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1 holds
b2 is dense(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 = the carrier of b2
holds b3 is dense(b1);
:: TEX_3:th 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
st b3 = the carrier of b2
holds b2 is dense(b1)
iff
b3 is dense(b1);
:: TEX_3:condreg 1
registration
let a1 be non empty TopSpace-like TopStruct;
cluster closed dense -> non proper (SubSpace of a1);
end;
:: TEX_3:condreg 2
registration
let a1 be non empty TopSpace-like TopStruct;
cluster proper dense -> non closed (SubSpace of a1);
end;
:: TEX_3:condreg 3
registration
let a1 be non empty TopSpace-like TopStruct;
cluster closed proper -> non dense (SubSpace of a1);
end;
:: TEX_3:exreg 1
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty strict TopSpace-like dense SubSpace of a1;
end;
:: TEX_3:th 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
st b2 is dense(b1)
holds ex b3 being non empty strict dense SubSpace of b1 st
b2 = the carrier of b3;
:: TEX_3:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty dense SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b3 = b4
holds b4 is dense(b2)
iff
b3 is dense(b1);
:: TEX_3:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being dense SubSpace of b1
for b3 being SubSpace of b1
st b2 is SubSpace of b3
holds b3 is dense(b1);
:: TEX_3:th 13
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty dense SubSpace of b1
for b3 being non empty SubSpace of b1
st b2 is SubSpace of b3
holds b2 is dense SubSpace of b3;
:: TEX_3:th 14
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty dense SubSpace of b1
for b3 being non empty dense SubSpace of b2 holds
b3 is non empty dense SubSpace of b1;
:: TEX_3:th 15
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
st b3 = TopStruct(#the carrier of b2,the topology of b2#)
holds b2 is dense SubSpace of b1
iff
b3 is dense SubSpace of b1;
:: TEX_3:attrnot 2 => TEX_3:attr 2
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be SubSpace of a1;
attr a2 is everywhere_dense means
for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
holds b1 is everywhere_dense(a1);
end;
:: TEX_3:dfs 2
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be SubSpace of a1;
To prove
a2 is everywhere_dense
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
holds b1 is everywhere_dense(a1);
:: TEX_3:def 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1 holds
b2 is everywhere_dense(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 = the carrier of b2
holds b3 is everywhere_dense(b1);
:: TEX_3:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
st b3 = the carrier of b2
holds b2 is everywhere_dense(b1)
iff
b3 is everywhere_dense(b1);
:: TEX_3:condreg 4
registration
let a1 be non empty TopSpace-like TopStruct;
cluster everywhere_dense -> dense (SubSpace of a1);
end;
:: TEX_3:condreg 5
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non dense -> non everywhere_dense (SubSpace of a1);
end;
:: TEX_3:condreg 6
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non proper -> everywhere_dense (SubSpace of a1);
end;
:: TEX_3:condreg 7
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non everywhere_dense -> proper (SubSpace of a1);
end;
:: TEX_3:exreg 2
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty strict TopSpace-like everywhere_dense SubSpace of a1;
end;
:: TEX_3:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
st b2 is everywhere_dense(b1)
holds ex b3 being non empty strict everywhere_dense SubSpace of b1 st
b2 = the carrier of b3;
:: TEX_3:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty everywhere_dense SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b3 = b4
holds b4 is everywhere_dense(b2)
iff
b3 is everywhere_dense(b1);
:: TEX_3:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being everywhere_dense SubSpace of b1
for b3 being SubSpace of b1
st b2 is SubSpace of b3
holds b3 is everywhere_dense(b1);
:: TEX_3:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty everywhere_dense SubSpace of b1
for b3 being non empty SubSpace of b1
st b2 is SubSpace of b3
holds b2 is everywhere_dense SubSpace of b3;
:: TEX_3:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty everywhere_dense SubSpace of b1
for b3 being non empty everywhere_dense SubSpace of b2 holds
b3 is everywhere_dense SubSpace of b1;
:: TEX_3:th 22
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
st b3 = TopStruct(#the carrier of b2,the topology of b2#)
holds b2 is everywhere_dense SubSpace of b1
iff
b3 is everywhere_dense SubSpace of b1;
:: TEX_3:condreg 8
registration
let a1 be non empty TopSpace-like TopStruct;
cluster open dense -> everywhere_dense (SubSpace of a1);
end;
:: TEX_3:condreg 9
registration
let a1 be non empty TopSpace-like TopStruct;
cluster dense non everywhere_dense -> non open (SubSpace of a1);
end;
:: TEX_3:condreg 10
registration
let a1 be non empty TopSpace-like TopStruct;
cluster open non everywhere_dense -> non dense (SubSpace of a1);
end;
:: TEX_3:exreg 3
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty strict TopSpace-like open dense SubSpace of a1;
end;
:: TEX_3:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
st b2 is dense(b1) & b2 is open(b1)
holds ex b3 being non empty strict open dense SubSpace of b1 st
b2 = the carrier of b3;
:: TEX_3:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1 holds
b2 is everywhere_dense(b1)
iff
ex b3 being strict open dense SubSpace of b1 st
b3 is SubSpace of b2;
:: TEX_3:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st (b2 is dense(b1) or b3 is dense(b1))
holds b2 union b3 is dense SubSpace of b1;
:: TEX_3:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st (b2 is everywhere_dense(b1) or b3 is everywhere_dense(b1))
holds b2 union b3 is everywhere_dense SubSpace of b1;
:: TEX_3:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st b2 is everywhere_dense(b1) & b3 is everywhere_dense(b1)
holds b2 meet b3 is everywhere_dense SubSpace of b1;
:: TEX_3:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st (b2 is everywhere_dense(b1) & b3 is dense(b1) or b2 is dense(b1) & b3 is everywhere_dense(b1))
holds b2 meet b3 is dense SubSpace of b1;
:: TEX_3:attrnot 3 => TEX_3:attr 3
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be SubSpace of a1;
attr a2 is boundary means
for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
holds b1 is boundary(a1);
end;
:: TEX_3:dfs 3
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be SubSpace of a1;
To prove
a2 is boundary
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
holds b1 is boundary(a1);
:: TEX_3:def 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1 holds
b2 is boundary(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 = the carrier of b2
holds b3 is boundary(b1);
:: TEX_3:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
st b3 = the carrier of b2
holds b2 is boundary(b1)
iff
b3 is boundary(b1);
:: TEX_3:condreg 11
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty open -> non boundary (SubSpace of a1);
end;
:: TEX_3:condreg 12
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty boundary -> non open (SubSpace of a1);
end;
:: TEX_3:condreg 13
registration
let a1 be non empty TopSpace-like TopStruct;
cluster everywhere_dense -> non boundary (SubSpace of a1);
end;
:: TEX_3:condreg 14
registration
let a1 be non empty TopSpace-like TopStruct;
cluster boundary -> non everywhere_dense (SubSpace of a1);
end;
:: TEX_3:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
st b2 is boundary(b1)
holds ex b3 being strict SubSpace of b1 st
b3 is boundary(b1) & b2 = the carrier of b3;
:: TEX_3:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being SubSpace of b1
st b2,b3 constitute_a_decomposition
holds b2 is dense(b1)
iff
b3 is boundary(b1);
:: TEX_3:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st b2,b3 constitute_a_decomposition
holds b2 is boundary(b1)
iff
b3 is dense(b1);
:: TEX_3:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
st b2 is boundary(b1)
for b3 being Element of bool the carrier of b1
st b3 c= the carrier of b2
holds b3 is boundary(b1);
:: TEX_3:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being SubSpace of b1
st b2 is boundary(b1) & b3 is SubSpace of b2
holds b3 is boundary(b1);
:: TEX_3:attrnot 4 => TEX_3:attr 4
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be SubSpace of a1;
attr a2 is nowhere_dense means
for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
holds b1 is nowhere_dense(a1);
end;
:: TEX_3:dfs 4
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be SubSpace of a1;
To prove
a2 is nowhere_dense
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
holds b1 is nowhere_dense(a1);
:: TEX_3:def 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1 holds
b2 is nowhere_dense(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 = the carrier of b2
holds b3 is nowhere_dense(b1);
:: TEX_3:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
st b3 = the carrier of b2
holds b2 is nowhere_dense(b1)
iff
b3 is nowhere_dense(b1);
:: TEX_3:condreg 15
registration
let a1 be non empty TopSpace-like TopStruct;
cluster nowhere_dense -> boundary (SubSpace of a1);
end;
:: TEX_3:condreg 16
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non boundary -> non nowhere_dense (SubSpace of a1);
end;
:: TEX_3:condreg 17
registration
let a1 be non empty TopSpace-like TopStruct;
cluster nowhere_dense -> non dense (SubSpace of a1);
end;
:: TEX_3:condreg 18
registration
let a1 be non empty TopSpace-like TopStruct;
cluster dense -> non nowhere_dense (SubSpace of a1);
end;
:: TEX_3:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
st b2 is nowhere_dense(b1)
holds ex b3 being strict SubSpace of b1 st
b3 is nowhere_dense(b1) & b2 = the carrier of b3;
:: TEX_3:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being SubSpace of b1
st b2,b3 constitute_a_decomposition
holds b2 is everywhere_dense(b1)
iff
b3 is nowhere_dense(b1);
:: TEX_3:th 38
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st b2,b3 constitute_a_decomposition
holds b2 is nowhere_dense(b1)
iff
b3 is everywhere_dense(b1);
:: TEX_3:th 39
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being SubSpace of b1
st b2 is nowhere_dense(b1)
for b3 being Element of bool the carrier of b1
st b3 c= the carrier of b2
holds b3 is nowhere_dense(b1);
:: TEX_3:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being SubSpace of b1
st b2 is nowhere_dense(b1) & b3 is SubSpace of b2
holds b3 is nowhere_dense(b1);
:: TEX_3:condreg 19
registration
let a1 be non empty TopSpace-like TopStruct;
cluster closed boundary -> nowhere_dense (SubSpace of a1);
end;
:: TEX_3:condreg 20
registration
let a1 be non empty TopSpace-like TopStruct;
cluster boundary non nowhere_dense -> non closed (SubSpace of a1);
end;
:: TEX_3:condreg 21
registration
let a1 be non empty TopSpace-like TopStruct;
cluster closed non nowhere_dense -> non boundary (SubSpace of a1);
end;
:: TEX_3:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
st b2 is boundary(b1) & b2 is closed(b1)
holds ex b3 being non empty strict closed SubSpace of b1 st
b3 is boundary(b1) & b2 = the carrier of b3;
:: TEX_3:th 42
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
b2 is nowhere_dense(b1)
iff
ex b3 being non empty strict closed SubSpace of b1 st
b3 is boundary(b1) & b2 is SubSpace of b3;
:: TEX_3:th 43
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st (b2 is boundary(b1) or b3 is boundary(b1)) & b2 meets b3
holds b2 meet b3 is boundary(b1);
:: TEX_3:th 44
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st b2 is nowhere_dense(b1) & b3 is nowhere_dense(b1)
holds b2 union b3 is nowhere_dense(b1);
:: TEX_3:th 45
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st (b2 is nowhere_dense(b1) & b3 is boundary(b1) or b2 is boundary(b1) & b3 is nowhere_dense(b1))
holds b2 union b3 is boundary(b1);
:: TEX_3:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
st (b2 is nowhere_dense(b1) or b3 is nowhere_dense(b1)) & b2 meets b3
holds b2 meet b3 is nowhere_dense(b1);
:: TEX_3:th 47
theorem
for b1 being non empty TopSpace-like TopStruct
st for b2 being SubSpace of b1 holds
b2 is not boundary(b1)
holds b1 is discrete;
:: TEX_3:th 48
theorem
for b1 being non empty non trivial TopSpace-like TopStruct
st for b2 being proper SubSpace of b1 holds
b2 is not dense(b1)
holds b1 is discrete;
:: TEX_3:condreg 22
registration
let a1 be non empty TopSpace-like discrete TopStruct;
cluster non empty -> non boundary (SubSpace of a1);
end;
:: TEX_3:condreg 23
registration
let a1 be non empty TopSpace-like discrete TopStruct;
cluster proper -> non dense (SubSpace of a1);
end;
:: TEX_3:condreg 24
registration
let a1 be non empty TopSpace-like discrete TopStruct;
cluster dense -> non proper (SubSpace of a1);
end;
:: TEX_3:exreg 4
registration
let a1 be non empty TopSpace-like discrete TopStruct;
cluster non empty strict TopSpace-like closed open discrete almost_discrete non boundary SubSpace of a1;
end;
:: TEX_3:exreg 5
registration
let a1 be non empty non trivial TopSpace-like discrete TopStruct;
cluster strict TopSpace-like closed open discrete almost_discrete non dense SubSpace of a1;
end;
:: TEX_3:th 49
theorem
for b1 being non empty TopSpace-like TopStruct
st ex b2 being non empty SubSpace of b1 st
b2 is boundary(b1)
holds b1 is not discrete;
:: TEX_3:th 50
theorem
for b1 being non empty TopSpace-like TopStruct
st ex b2 being non empty SubSpace of b1 st
b2 is dense(b1) & b2 is proper(b1)
holds b1 is not discrete;
:: TEX_3:exreg 6
registration
let a1 be non empty TopSpace-like non discrete TopStruct;
cluster non empty strict TopSpace-like boundary SubSpace of a1;
end;
:: TEX_3:exreg 7
registration
let a1 be non empty TopSpace-like non discrete TopStruct;
cluster non empty strict TopSpace-like proper dense SubSpace of a1;
end;
:: TEX_3:th 51
theorem
for b1 being non empty TopSpace-like non discrete TopStruct
for b2 being non empty Element of bool the carrier of b1
st b2 is boundary(b1)
holds ex b3 being strict boundary SubSpace of b1 st
b2 = the carrier of b3;
:: TEX_3:th 52
theorem
for b1 being non empty TopSpace-like non discrete TopStruct
for b2 being non empty proper Element of bool the carrier of b1
st b2 is dense(b1)
holds ex b3 being strict proper dense SubSpace of b1 st
b2 = the carrier of b3;
:: TEX_3:th 53
theorem
for b1 being non empty TopSpace-like non discrete TopStruct
for b2 being non empty boundary SubSpace of b1 holds
ex b3 being non empty strict proper dense SubSpace of b1 st
b2,b3 constitute_a_decomposition;
:: TEX_3:th 54
theorem
for b1 being non empty TopSpace-like non discrete TopStruct
for b2 being non empty proper dense SubSpace of b1 holds
ex b3 being non empty strict boundary SubSpace of b1 st
b2,b3 constitute_a_decomposition;
:: TEX_3:th 55
theorem
for b1 being non empty TopSpace-like non discrete TopStruct
for b2, b3 being non empty TopSpace-like TopStruct
st b3 = TopStruct(#the carrier of b2,the topology of b2#)
holds b2 is boundary SubSpace of b1
iff
b3 is boundary SubSpace of b1;
:: TEX_3:th 56
theorem
for b1 being non empty TopSpace-like TopStruct
st for b2 being SubSpace of b1 holds
b2 is not nowhere_dense(b1)
holds b1 is almost_discrete;
:: TEX_3:th 57
theorem
for b1 being non empty non trivial TopSpace-like TopStruct
st for b2 being proper SubSpace of b1 holds
b2 is not everywhere_dense(b1)
holds b1 is almost_discrete;
:: TEX_3:condreg 25
registration
let a1 be non empty TopSpace-like almost_discrete TopStruct;
cluster non empty -> non nowhere_dense (SubSpace of a1);
end;
:: TEX_3:condreg 26
registration
let a1 be non empty TopSpace-like almost_discrete TopStruct;
cluster proper -> non everywhere_dense (SubSpace of a1);
end;
:: TEX_3:condreg 27
registration
let a1 be non empty TopSpace-like almost_discrete TopStruct;
cluster everywhere_dense -> non proper (SubSpace of a1);
end;
:: TEX_3:condreg 28
registration
let a1 be non empty TopSpace-like almost_discrete TopStruct;
cluster non empty boundary -> non closed (SubSpace of a1);
end;
:: TEX_3:condreg 29
registration
let a1 be non empty TopSpace-like almost_discrete TopStruct;
cluster non empty closed -> non boundary (SubSpace of a1);
end;
:: TEX_3:condreg 30
registration
let a1 be non empty TopSpace-like almost_discrete TopStruct;
cluster proper dense -> non open (SubSpace of a1);
end;
:: TEX_3:condreg 31
registration
let a1 be non empty TopSpace-like almost_discrete TopStruct;
cluster open dense -> non proper (SubSpace of a1);
end;
:: TEX_3:condreg 32
registration
let a1 be non empty TopSpace-like almost_discrete TopStruct;
cluster open proper -> non dense (SubSpace of a1);
end;
:: TEX_3:exreg 8
registration
let a1 be non empty TopSpace-like almost_discrete TopStruct;
cluster non empty strict TopSpace-like non nowhere_dense SubSpace of a1;
end;
:: TEX_3:exreg 9
registration
let a1 be non empty non trivial TopSpace-like almost_discrete TopStruct;
cluster strict TopSpace-like non everywhere_dense SubSpace of a1;
end;
:: TEX_3:th 58
theorem
for b1 being non empty TopSpace-like TopStruct
st ex b2 being non empty SubSpace of b1 st
b2 is nowhere_dense(b1)
holds b1 is not almost_discrete;
:: TEX_3:th 59
theorem
for b1 being non empty TopSpace-like TopStruct
st ex b2 being non empty SubSpace of b1 st
b2 is boundary(b1) & b2 is closed(b1)
holds b1 is not almost_discrete;
:: TEX_3:th 60
theorem
for b1 being non empty TopSpace-like TopStruct
st ex b2 being non empty SubSpace of b1 st
b2 is everywhere_dense(b1) & b2 is proper(b1)
holds b1 is not almost_discrete;
:: TEX_3:th 61
theorem
for b1 being non empty TopSpace-like TopStruct
st ex b2 being non empty SubSpace of b1 st
b2 is dense(b1) & b2 is open(b1) & b2 is proper(b1)
holds b1 is not almost_discrete;
:: TEX_3:exreg 10
registration
let a1 be non empty TopSpace-like non almost_discrete TopStruct;
cluster non empty strict TopSpace-like nowhere_dense SubSpace of a1;
end;
:: TEX_3:exreg 11
registration
let a1 be non empty TopSpace-like non almost_discrete TopStruct;
cluster non empty strict TopSpace-like proper everywhere_dense SubSpace of a1;
end;
:: TEX_3:th 62
theorem
for b1 being non empty TopSpace-like non almost_discrete TopStruct
for b2 being non empty Element of bool the carrier of b1
st b2 is nowhere_dense(b1)
holds ex b3 being non empty strict nowhere_dense SubSpace of b1 st
b2 = the carrier of b3;
:: TEX_3:th 63
theorem
for b1 being non empty TopSpace-like non almost_discrete TopStruct
for b2 being non empty proper Element of bool the carrier of b1
st b2 is everywhere_dense(b1)
holds ex b3 being strict proper everywhere_dense SubSpace of b1 st
b2 = the carrier of b3;
:: TEX_3:th 64
theorem
for b1 being non empty TopSpace-like non almost_discrete TopStruct
for b2 being non empty nowhere_dense SubSpace of b1 holds
ex b3 being non empty strict proper everywhere_dense SubSpace of b1 st
b2,b3 constitute_a_decomposition;
:: TEX_3:th 65
theorem
for b1 being non empty TopSpace-like non almost_discrete TopStruct
for b2 being non empty proper everywhere_dense SubSpace of b1 holds
ex b3 being non empty strict nowhere_dense SubSpace of b1 st
b2,b3 constitute_a_decomposition;
:: TEX_3:th 66
theorem
for b1 being non empty TopSpace-like non almost_discrete TopStruct
for b2, b3 being non empty TopSpace-like TopStruct
st b3 = TopStruct(#the carrier of b2,the topology of b2#)
holds b2 is nowhere_dense SubSpace of b1
iff
b3 is nowhere_dense SubSpace of b1;
:: TEX_3:exreg 12
registration
let a1 be non empty TopSpace-like non almost_discrete TopStruct;
cluster non empty strict TopSpace-like closed boundary SubSpace of a1;
end;
:: TEX_3:exreg 13
registration
let a1 be non empty TopSpace-like non almost_discrete TopStruct;
cluster non empty strict TopSpace-like open proper dense SubSpace of a1;
end;
:: TEX_3:th 67
theorem
for b1 being non empty TopSpace-like non almost_discrete TopStruct
for b2 being non empty Element of bool the carrier of b1
st b2 is boundary(b1) & b2 is closed(b1)
holds ex b3 being non empty strict closed boundary SubSpace of b1 st
b2 = the carrier of b3;
:: TEX_3:th 68
theorem
for b1 being non empty TopSpace-like non almost_discrete TopStruct
for b2 being non empty proper Element of bool the carrier of b1
st b2 is dense(b1) & b2 is open(b1)
holds ex b3 being strict open proper dense SubSpace of b1 st
b2 = the carrier of b3;
:: TEX_3:th 69
theorem
for b1 being non empty TopSpace-like non almost_discrete TopStruct
for b2 being non empty closed boundary SubSpace of b1 holds
ex b3 being non empty strict open proper dense SubSpace of b1 st
b2,b3 constitute_a_decomposition;
:: TEX_3:th 70
theorem
for b1 being non empty TopSpace-like non almost_discrete TopStruct
for b2 being non empty open proper dense SubSpace of b1 holds
ex b3 being non empty strict closed boundary SubSpace of b1 st
b2,b3 constitute_a_decomposition;
:: TEX_3:th 71
theorem
for b1 being non empty TopSpace-like non almost_discrete TopStruct
for b2 being non empty SubSpace of b1 holds
b2 is nowhere_dense(b1)
iff
ex b3 being non empty strict closed boundary SubSpace of b1 st
b2 is SubSpace of b3;
:: TEX_3:th 72
theorem
for b1 being non empty TopSpace-like non almost_discrete TopStruct
for b2 being non empty nowhere_dense SubSpace of b1
st (b2 is boundary(b1) implies b2 is not closed(b1))
holds ex b3 being non empty strict proper everywhere_dense SubSpace of b1 st
ex b4 being non empty strict closed boundary SubSpace of b1 st
b3 meet b4 = TopStruct(#the carrier of b2,the topology of b2#) &
b3 union b4 = TopStruct(#the carrier of b1,the topology of b1#);
:: TEX_3:th 73
theorem
for b1 being non empty TopSpace-like non almost_discrete TopStruct
for b2 being non empty everywhere_dense SubSpace of b1
st (b2 is dense(b1) implies b2 is not open(b1))
holds ex b3 being non empty strict open proper dense SubSpace of b1 st
ex b4 being non empty strict nowhere_dense SubSpace of b1 st
b3 misses b4 &
b3 union b4 = TopStruct(#the carrier of b2,the topology of b2#);
:: TEX_3:th 74
theorem
for b1 being non empty TopSpace-like non almost_discrete TopStruct
for b2 being non empty nowhere_dense SubSpace of b1 holds
ex b3 being non empty strict open proper dense SubSpace of b1 st
ex b4 being non empty strict closed boundary SubSpace of b1 st
b3,b4 constitute_a_decomposition & b2 is SubSpace of b4;
:: TEX_3:th 75
theorem
for b1 being non empty TopSpace-like non almost_discrete TopStruct
for b2 being proper everywhere_dense SubSpace of b1 holds
ex b3 being strict open proper dense SubSpace of b1 st
ex b4 being strict closed boundary SubSpace of b1 st
b3,b4 constitute_a_decomposition & b3 is SubSpace of b2;