I'm studying covering maps and homotopy lifting and I would like to clarify a few things which my lecture notes doesn't seem to make clear.
A lemma in my lecture notes says:
Let $p: \tilde Y \to Y$ be a covering map, and let $f:X \to Y$ be a continuous map, with $X$ connected. Suppose that $\tilde f_1 : X \to \tilde Y$ and $\tilde f_2 : X \to \tilde Y$ are lifts of $f$. If $\tilde f_1(X) = \tilde f_2(x)$ for some point $x \in X$, then $\tilde f_1$ and $\tilde f_2$ agree everywhere.
But it true given a covering map $p: \tilde Y \to Y$, and given a continuous map $f:X \to Y$, you can always find a lift of $f$ to $\tilde Y$?
I know this is true for a path in $Y$ but does this generalize?