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I have a homework as follows:

Find connected space $X$ such that all continuous real-valued functions defined on $X$ is constant!

Please help me to find a such space

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  • $\begingroup$ What’s the absolutely simplest space that you can think of — a very small space? $\endgroup$ Commented Dec 9, 2014 at 5:10
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    $\begingroup$ For a less trivial example, you could try proving that an infinite set with the cofinite topology has the desired properties; it’s very straightforward. $\endgroup$ Commented Dec 9, 2014 at 5:19
  • $\begingroup$ singleton set is too trivial example. $\endgroup$
    – flourence
    Commented Dec 9, 2014 at 5:27
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    $\begingroup$ In that case, you should specify that in the body of your question. $\endgroup$
    – Nick
    Commented Dec 9, 2014 at 5:30

3 Answers 3

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Since you want an example less trivial than the one-point space, I’ll post my comment as an answer: try showing that an infinite set with the cofinite topology has the desired properties.

It’s not Hausdorff; Hausdorff examples are much harder to find. There are even $T_3$ examples, though they’re pretty complicated; if you’re interested, you can learn more about them from the answers to this question.

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a single point!

more generally, take any set and take the topology to be the empty set and the whole space

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If you consider X={p} just one point! Every function from X to the reals is necessarily 1) continuous, and 2) constant !

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