For which subspace $X \subseteq \mathrm{R}$ with usual topology and with $\{0,1\}\subseteq X$, will a continuous function $ f: X \to \{0,1\}$ satisfying $f(0)=0$ and $f(1)=1$ exist?

  1. $X= [0,1]$
  2. $X= [-1,1]$
  3. $X= \mathrm{R}$
  4. $[0,1] \not \subset X$
  • 2
    $\begingroup$ Tip: Let $X$ be a connected topological space and $f: X \rightarrow \{0,1\}$ a continuous function, then $f$ is constant. $\endgroup$ – G. Chiusole Feb 14 at 8:09

Since for the first three options the domains are connected but the range is not connected the only possible case is the fourth option which $[0,1] \not \subset X$


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