If we have a covering map $p:\tilde{X}\to X$ such that $\tilde{X}$ is connected and $X$ is path connected is true that $\tilde{X}$ is path-connected?

Question

if we have a covering map $p:\tilde{X}\to X$ such that $\tilde{X}$ is connected and $X$ is path connected is true that $\tilde{X}$ is path-connected?

I was trying use the path lifting property i take $\tilde{x}, \tilde{y} \in \tilde{X}$ and then how $X$ is path connected there exist a path from $x$ to $y$ and by lifting property there exist a lift such that starts in $\tilde{x}$ and the endpopint must be in the fiber of $y$ but icant conclude nothing because i dont know if its true or not thanks

Answer 1

is true that $\tilde{X}$ is path-connected?

No, not necessarily.

Consider the Warsaw circle as $X$:

and insert another topologist's sine curve inside:

which will be our $\tilde{X}$.

This is a covering of degree $2$ (each "semi Warsaw circle" is mapped onto entire Warsaw circle), with $X$ path-connected but $\tilde{X}$ only connected.