topological space

\(\underline{Def:}\)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)
\(\bullet\)Different distance funcs might determine the same topology