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(1) $X= (0,1]$ and $Y = [0,1]$;

(2) $X = [0,1]$ and $Y$ is the topologist's sine curve

(3) $X = [0,1] \cup [2,3]$ and $Y = X \times X$

I believe there is a continuous function for the first one.

I know there doesn't exist a continuous function for the second one, because $X$ is path connected and the topologist's sine curve is not. As well as, $f(0)$ doesn't exist in $Y$.

I feel like there is no continuous function for (3) because of the disconnection; however, I could be wrong.

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  • 3
    $\begingroup$ Count connected components. $\endgroup$ – Daniel Fischer Apr 25 '14 at 18:44
  • $\begingroup$ What topology do the sets have? A function $\mathrm{f} : X \to Y$ is called continuous if the pre-image of each open set in $Y$ is an open set in $X$. To know if this is true, or not, you need to know which sets are open. You need to state the topologies that you are using on each choice of $X$ and $Y$. For example, if $X$ is given the discreet topology then every function is continuous. $\endgroup$ – Fly by Night Apr 25 '14 at 19:00
  • $\begingroup$ Well, you can take $f(x)=|2x-1|$ (or some similar piecewise linear function) to get a surjective continuous map from $(0,1]$ onto $[0,1]$. $\endgroup$ – Martin Sleziak Apr 25 '14 at 20:38
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These problems all use variations on the idea of connectivity. Let me give you some hints.

  1. $Y$ has the property that if you remove two certain points (namely $0$ and $1$), it is still connected. Does $X$ has this property?

  2. Are $X$ and $Y$ path-connected?

  3. How many connected components does $X$ have? What about $Y$?

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  • $\begingroup$ (2) I know that X is path-connected, but Y is not. $\endgroup$ – user114634 Apr 26 '14 at 18:32
  • $\begingroup$ That's right. So if there were a surjective continuous function $X\to Y$ we could take any path in $X$ and map it over to get a path in $Y$. Do you see a contradiction? $\endgroup$ – user134824 Apr 26 '14 at 18:36
  • $\begingroup$ There's no f(0) in Y. $\endgroup$ – user114634 Apr 26 '14 at 18:39
  • $\begingroup$ (3) X obviously have 2 connect components, and I believe that Y has 3 connected components. So there can't be a continuous function. $\endgroup$ – user114634 Apr 26 '14 at 19:30

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