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I started general topology recently, and a partner of mine made a example of two different topologies generated by the same basis, so I'd like to know if that is possible, and if it is not, to see a proof of it.

joseph morrison
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    This is not possible. It would be most helpful to actually see the details of this partner's example. – Randall Apr 14 '21 at 18:33
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    Just consider an arbitrary open set in one of the topologies, represent it using the basis, and then it is also an element of the other topology by definition of a basis. – Jürgen Sukumaran Apr 14 '21 at 18:34
  • Not if they are topologies on the same set. If they are topologies on different sets... well, one usually speaks of a basis in context (as, "a subset of the power set of $X$"), so one would not normally call them "the same basis". – Arturo Magidin Apr 14 '21 at 18:39
  • Obviously if you are saying that your partner did it, then it is possible, so then there is nothing to ask. – Mirko Apr 15 '21 at 02:33

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