$(X,\tau_X)$ is a path connected, then is there a continuous function from $(X,\tau_X)\rightarrow(\mathbb{R},\tau_E)$ with , $f(x)=0$ and $f(y)=1$

Question

I think the answer to this question is yes there exists. My main reasoning is that since there is a continuous path $\gamma: ([0,1],\tau_E)\rightarrow (X,\tau_X)$, with $\gamma(0)=x$ and $\gamma(1)=y$ I can consider $\gamma^{-1}$ but I don't know how to prove that this would be continuous, and I end up doubting myself.

Any help is appreciated thank you!

Answer 1

You will need some extra hypotheses: Suppose $X$ has the trivial topology. Then it is automatically path connected and every continuous map $f:X\to \mathbb{R}$ must be constant.