(adsbygoogle = window.adsbygoogle || []).push({}); So, if f could be any function satisfying f(x)=0 and f(y)=1, then is this specifying a "function space"? And are there functionals that can be identified on that space? friend said: ↑Since a continuous function, f, must have a continous domain, and since every point in the topology can be considered a closed set which must be accommodated in definition 4), it seems that a completely regular space is path connected from any point to any other point. Is that enough to specify a manifold? Or do you also need the property of being Hausdorff?

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The definition in 4) states, "then there is a continuous function f..." So the question is whether there is anything else that specifies what f can be? Does that mean we are free to invent any functions we like from any point to any other point?

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# Completely regular spaces

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