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Let

  • $X$ be a set, and
  • $T_1,T_2$ be a pair of topologies on $X$.

Assume that, for all $f:X → X$, $f$ is continuous wrt $T_1$ iff $f$ is continuous wrt $T_2$. Must $T_1=T_2$?

If so, I'd like a proof and if not, a counterexample.

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  • $\begingroup$ I think you've mis-written your question. Is $f$ supposed to be one fixed map, or is the condition "cts w.r.t. $T_1$ iff cts w.r.t. $T_2$" supposed to be true for all $f$? I'd suggest removing your second bullet point. $\endgroup$
    – JonathanZ
    Jan 16, 2021 at 20:44
  • $\begingroup$ @JonathanZsupportsMonicaC, I have miss written in, however not in the way you suggested. I don't want to delete the question as people have put effort into answering it, so I'll just post the correct version as a new question. $\endgroup$
    – Mathew
    Jan 16, 2021 at 20:57

1 Answer 1

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If one is talking about functions $f:X → X$, I do not think it is true: consider a two-pointed space with

  • the trivial topology,
  • the power set topology.
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