(underline{Def:})A topology space
(mathcal{X}=(underline{X},eth_{x}))consists of a set (underline{X}),called the underlying space of (mathcal{X}) ,and a family (eth_{x})of subsets of (mathcal{X})(ie.(eth_{x}subset P(underline{X})))
(P(underline{X}))means the power set of (underline{X})
s.t.:(1):(underline{X}) and (varnothing in eth_{x})
(2):(U_{alpha}in eth_{x}(alpha in A) Rightarrow)
(cup_{alpha in A}U_{alpha} in eth_{x})
(3).(U,U^{prime}in eth_{x} Rightarrow U cap U^{prime} in eth_{x})
(eth_{x}) is called a topology(topological structure) on (underline{X})
(underline{Convention:})We usually use (mathcal{X}) to indicate the set (underline{X})and omit the subscript (x) in (eth_{x}) by saying "a topological space((X,eth))"
(underline{Examples:})(1)metric space:
((X,d) looparrowright(X,eth_{d}))(open sets induced by d)
(ullet)Different distance funcs might determine the same topology