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.
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.