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How must I prove this problem? Let $X$ be an infinite set and let $T$ a topology in $X$ in which all infinite subsets of $X$ are open.

Prove: $T$ is a discrete topology in $X$.

badet
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  • Welcome to MSE! Do you have any thoughts and can share what you have tried to help responders? Regards – Amzoti Jul 09 '13 at 04:37

1 Answers1

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HINT: Since $X$ is infinite, there is an infinite $A\subseteq X$ such that $X\setminus A$ is also infinite. For convenience let $B=X\setminus A$. For each $x\in A$ consider the open sets $A$ and $\{x\}\cup B$. Do something very similar to handle the points of $B$.

Brian M. Scott
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  • Thank you for your hint. But can you give methe complete proof of this problem? I am a newbie in proving and I find difficulty in formulating a proof. This is one of the problems that interests me when I tried to prove the supplemetary problems in Schaum's outline Book. – badet Jul 09 '13 at 04:21
  • @badet: Let’s see first if I can help you to work out most of it yourself. Let $x\in A$; do you see why $A$ and ${x}\cup B$ are open sets containing $x$? And what is the intersection of these two open sets? – Brian M. Scott Jul 09 '13 at 04:24
  • the union of a singleton set containing x and any other set still contains x and since B is infinite then {x}∪B is n open set containing x. The intersection is x. – badet Jul 09 '13 at 04:37
  • @badet: Small correction: the intersection is the set ${x}$, not the point $x$. But otherwise you’re right. And the intersection of two open sets is always open, so ${x}$ is an open set. $x$ was any point of $A$, and you can do something very similar to show that ${x}$ is open if $x\in B$. Once you’ve done that, you know that ${x}$ is open for every $x\in X$. That’s one definition of the discrete topology. If you were given a different definition, you should try to see why it’s equivalent to this one. – Brian M. Scott Jul 09 '13 at 04:42
  • each time I read a proof, I find it fascinating how people can do it which much ease. how I wish i can do the same. – badet Jul 09 '13 at 04:43
  • @badet: Partly it’s practice. Partly it’s paying attention when reading proofs and picking up the style and some fairly common general patterns of thought. I won’t deny that it comes more easily to some people than to others, but I’ve had students improve enormously over the course of a semester. – Brian M. Scott Jul 09 '13 at 04:45
  • Thank you so much sir. I really understand better the concepts once I read its proof. Though I still love to read the proof of this problem from a pro. A million thanks, Sir. I hope to learn a lot from you through this site. – badet Jul 09 '13 at 04:50
  • @badet: You’re very welcome. Good luck! – Brian M. Scott Jul 09 '13 at 04:53