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Hi I've got a problem on topology.

Let $\mathcal T$ be the class of subsets of positive integers consisting null set and all subsets of positive integers of the form $E_n =\{n,n+1,n+2,n+3,\dots\}$ with $n$ element of positive integers.

  1. Show that $\mathcal T$ is a topology on the set of positive integers.
  2. List the open sets containing the positive integer $G$.

Could you please help me?

  • 0
    Since you are new to this site, you could maybe read this: [How to ask a homework question.](http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question) I wrote this comment because the question sounds homework-like.2012-08-03

1 Answers 1

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  1. You have to check that

    • the null set is in $\mathcal T$, which is assumed;
    • $\Bbb N^*$, the whole set, is in $\mathcal T$, which can be seen taking a particular value of $n$;
    • the intersection of two elements of $\mathcal T$ still is in $\mathcal T$. It's clear when one of them is the null set, otherwise, writing $S_n:=\{k\in\Bbb N^*,k\geq n\}$, we have $S_n\cap S_m=S_{f(m,n)}$, where you have to find $f(m,n)$;
    • an arbitrary union of elements $S_i,i\in I$ of $\mathcal T$ still is in $\mathcal T$. You can assume that none of these elements is the null set. Then show that $\bigcup_{i\in I} S_i=S_{\min_{j\in J}j}$.
  2. The open sets are given; surely the null set won't work. Can you see which $S_j$ will work?