let Y be TopStruct ; :: thesis: for A being Subset of Y st A is discrete holds
for x being Point of Y st x in A holds
ex G being Subset of Y st
( G is open & A /\ G = {x} )
let A be Subset of Y; :: thesis: ( A is discrete implies for x being Point of Y st x in A holds
ex G being Subset of Y st
( G is open & A /\ G = {x} ) )
assume A1: A is discrete ; :: thesis: for x being Point of Y st x in A holds
ex G being Subset of Y st
( G is open & A /\ G = {x} )
let x be Point of Y; :: thesis: ( x in A implies ex G being Subset of Y st
( G is open & A /\ G = {x} ) )
assume A2: x in A ; :: thesis: ex G being Subset of Y st
( G is open & A /\ G = {x} )
then reconsider Y' = Y as non empty TopStruct ;
reconsider A' = A as Subset of Y' ;
reconsider B = {x} as Subset of Y' by ZFMISC_1:37;
{x} c= A' by A2, ZFMISC_1:37;
then consider G being Subset of Y' such that
A3: ( G is open & A' /\ G = B ) by A1, Def5;
reconsider G = G as Subset of Y ;
take G ; :: thesis: ( G is open & A /\ G = {x} )
thus ( G is open & A /\ G = {x} ) by A3; :: thesis: verum