jimlucky的部落格

Define a Basis for a topology on X is
collection subsets of X s.t
1.for any x belong to X,we can find a B1 s.t x belong to B1
2.for any x belong to B1 intersection B2, B1 B2 belong to B
then there exists B3 s.t x belong to B3,
B3 contains in B1 intersection B2

Basis can generate a Topology for X
(is the T unique ?)
but is B a Topology?
consider X=R
S={(a,b)| a,b belong to R}
and S`={(c,d)| c,d belong to Q }
for any elt in S (a,b)
because Q dense in R,
we can find a two Cauchy sequence converges to a and b
a1 a2 a3....---> a (ai belong to Q )
b1 b2 b3....---> b (bi belong to Q )
then we find elt (a1 ,b1)....(an ,bn) in S`
and union of these elt can generate (a,b)
so S` can generate S.
but it`s also mean S` isn`t a Topolopy.
( union of Bi doesn`t in B )