Topology space

\(\underline{Def:}\)A topology space
\(X=(\underline{X},\eth_{x})\)consists of a set \(\underline{X}\),called the underlying space of \(X\) ,and a family \(\eth_{x}\)of subsets of \(X\)(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 \(\underline{X}\)
Convention: