2.I.4E2 . \mathrm{I} . 4 \mathrm{E} \quad

Further Analysis | Part IB, 2003

Let τ1\tau_{1} be the collection of all subsets ANA \subset \mathbb{N} such that A=A=\emptyset or N\A\mathbb{N} \backslash A is finite. Let τ2\tau_{2} be the collection of all subsets of N\mathbb{N} of the form In={n,n+1,n+2,}I_{n}=\{n, n+1, n+2, \ldots\}, together with the empty set. Prove that τ1\tau_{1} and τ2\tau_{2} are both topologies on N\mathbb{N}.

Show that a function ff from the topological space (N,τ1)\left(\mathbb{N}, \tau_{1}\right) to the topological space (N,τ2)\left(\mathbb{N}, \tau_{2}\right) is continuous if and only if one of the following alternatives holds:

(i) f(n)f(n) \rightarrow \infty as nn \rightarrow \infty;

(ii) there exists NNN \in \mathbb{N} such that f(n)=Nf(n)=N for all but finitely many nn and f(n)Nf(n) \leqslant N for all nn.

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