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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What’s the absolutely simplest space that you can think of — a very small space? – Brian M. Scott Dec 09 '14 at 05:10
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3For a less trivial example, you could try proving that an infinite set with the cofinite topology has the desired properties; it’s very straightforward. – Brian M. Scott Dec 09 '14 at 05:19
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singleton set is too trivial example. – flourence Dec 09 '14 at 05:27
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2In that case, you should specify that in the body of your question. – Nick Dec 09 '14 at 05:30
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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.
Brian M. Scott
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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
Mark Joshi
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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 !
Idris Addou
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