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Consider the topological space $X = \{0, 1\}, τ = \{∅, X, \{0\}\}$

Show that X is path-connected.

Let $t ∈]0, 1]$, I try to prove that the path $γ : [0, 1] → \{0, 1\}$ sending $[0, t[$ over $0$ and $[t, 1]$ over 1 is continuous.

But for the open element $X$, we have $γ^{-1}(X)=[0,1]$ which is not open for the standard topology on $\mathbb{R}$?

Thanks for your help.

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    $\begingroup$ You claim that $[0,1]$ is not closed for the standard topology on $\mathbb R$. Sure about that? $\endgroup$ – José Carlos Santos Jan 24 at 16:05
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    $\begingroup$ I think it's a typo and he meant "open in the standard topology". Note that it should be open in $[0,1]$, not in $\mathbb{R}$. And indeed $[0,1]$ is open in $[0,1]$. $\endgroup$ – freakish Jan 24 at 16:07
  • $\begingroup$ I see, I was considering $[0,1]\subset \mathbb{R}$. Thank you for your hint! $\endgroup$ – PerelMan Jan 24 at 16:08
  • $\begingroup$ @JoséCarlosSantos: Thank you, I meant not open $\endgroup$ – PerelMan Jan 24 at 19:15
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Just define $\gamma(t)= 0$ for all $t \in [0,1)$ and $\gamma(1)=1$

Then $\gamma^{-1}[\emptyset]=\emptyset=\emptyset \cap [0,1]$, $\gamma^{-1}[\{0\}] = [0,1)=(-1,1) \cap [0,1]$ and $\gamma^{-1}[X]=[0,1]=\mathbb{R} \cap [0,1]$.

All three inverse images of open sets are open in $[0,1]$ in the subspace topology as witnessed by the intersection representations.

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