Given metric spaces(adsbygoogle = window.adsbygoogle || []).push({});

[tex](X, d_X), (Y, d_Y)[/tex],

and subsets

[tex]\{ x_1, ..., x_n \}, \{ y_1, ..., y_n \}[/tex]

of X and Y respectively, if I define a function that send [tex]x_i[/tex] to [tex]y_i[/tex], what are the minimal restrictions we need on X and Y in order for there to exist an extension of that map to a continuous function defined on all of X?

Certainly you'd need some relationship between the connected components of both. So for simplicity, also assume each of the metric spaces are (path) connected. Also assume X and Y are compact.

If you make any assumptions on Y that will allow you to apply Uryshon's lemma or Tietze extension theorem, then it is not very interesting. I'm curious about the minimal set of restrictions on Y needed to find this extension. (Perhaps I'm missing some elementary theorem that would give my answer?).

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# Criteria for extension to cts fn between metric spaces?

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