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Topological Space**

A 'topological space' \((X,\mathcal{T}_X)\) consists of a set \(X\) (the 'space') and a set \(\mathcal{T}_X\subseteq\mathcal{P}(X)\) of subsets of X, the 'topology' on \(X\), the elements of which are called the 'open subsets' of \(X\). \(\mathcal{T}_X\) is required to satisfy the following: The total space and the empty set must be open: \(X,\emptyset\in\mathcal{T}_X\); The union of any collection of open sets must be open: \(\forall\mathcal{U}\subseteq\mathcal{T}_n:\cup\mathcal{U}\in\mathcal{T}_X\); The intersection of two (and hence any finite number of) open sets must be open: \(\forall U_1,U_2\in\mathcal{T}_X:U_1\cap U_2\in\mathcal{T}_X\).