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I want to prove that the only continuous function from $R$ (with usual topology) to $R$ ( the Sorgenfrey line's $R_l$ Lower limit topology) is constant function.

My attempt is using the the properties of connected space. BOWC, assume there is continuous function which are not constant and surjective Since $f(R)$ is connected with more than one points. Then, it must be interval and $f(R)$ as subspace of $R_l$. I can not complete. I am asking wether or not my solution right.

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  • $\begingroup$ You're on the right track. Now you need to show that if $Y$ is a subspace of $\Bbb R_\ell$ with more than one point, $Y$ is not connected. $\endgroup$ – Brian M. Scott Aug 2 '16 at 17:19
  • $\begingroup$ This step that I have trouble to explain it could you give hint to finish it. $\endgroup$ – Gob Aug 2 '16 at 22:10
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HINT: You need to show that if $Y$ is a subspace of $\Bbb R_\ell$ with more than one point, then $Y$ is not connected. To do this, let $Y$ be such a subspace, and let $x,y\in Y$ with $x<y$. Let

$$L=\{z\in Y:z<y\}$$

and

$$R=\{z\in Y:z\ge y\}\;.$$

Show that $Y=L\cup R$, $L\cap R=\varnothing$, and $L$ and $R$ are both open in the subspace $Y$.

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  • $\begingroup$ Thank you it is clear right now $\endgroup$ – Gob Aug 3 '16 at 4:21
  • $\begingroup$ @Gob: You're welcome. $\endgroup$ – Brian M. Scott Aug 3 '16 at 7:17

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