I'm reading "Complex Made Simple" by David C. Ullrich and here i have a problem with the proof of a theorem:
TheoremSuppose that $p : X \to Y$ is a covering map. If $\gamma : [0,1] \to Y$ is continuous, $x_0 \in X$ and $p(x_0) = \gamma(0)$ then there exists a unique continuous function $\tilde{\gamma} : [0,1] \to X$ such that $\tilde{\gamma}(0) = x_0$ and $p \circ \tilde{\gamma} = \gamma$.
This is the proof that author gives:
To prove the existence, let $A$ be the set of $t_0 \in [0,1]$ such that there exists a continuous $\tilde{\gamma} : [0, t_0] \to X$ with $\tilde{\gamma(0)} = x_0$ and $p(\tilde{\gamma}(t)) = \gamma(t)$ for all $t \in [0, t_0]$. We need to show that $A$ is both open and closed.
I don't know how to show that $A$ is both open and closed, and as a mention I never took a Algebraic Topology class before. I tried to read some books, but can't understand very good what happens there and in the book mentioned by the author (A Basic Course in Algebraic Topology by William S. Massey) I don't find this proof for existence (that one from the book I don't understand). Thanks!