unique lifing property

Question

I'm studying Hatcher's book by myslef. I am confused about the proof of Proposition 1.34 on page 62.

Given a covering space $p:\overline X → X $ and a map $f:Y → X$, if two lifts $\overline f_1 ,\overline f_2 :Y → \overline X $ of $f$ agree at one point of $\ Y $and $\ Y$ is connected, then$\overline f_1$ and $\overline f_2$ agree on all of $ \ Y .$

I understand the ideal of the proof. But the last step is not clear for me. He said that the set of points where $\overline f_1$ and $\overline f_2$ agree is both open and closed in $Y$. The set open is clearly, but why is closed. I am really don't know the reason.

Answer 1

You say you understand why the set of points they agree on is open. By the same reason, the set of points they disagree on is also open. So the complement (i.e., where they agree) is closed.

Answer 2

Okay, let us denote $$\mathscr{A}=\{y\in Y: \tilde{f_1}(y)=\tilde{f_2}(y)\}.$$ Hatcher proved that if $\tilde{f_1}(y)=\tilde{f_1}(y)$, then there is a neighborhood $N$ in $Y$ containing $y$ such that $\tilde{f_1}=\tilde{f_2}$ in $N$. But what is a neighborhood of $y$? It is a set containing a smaller open set $U_y\ni y$. Thus for every $y\in \mathscr{A}$, we have that $U_y\subset \mathscr{A}$. So we have $\mathscr{A}=\bigcup_{y\in \mathscr{A}} U_y$, which is a union of open sets, hence open.

The exact same argument shows that the complement of $\mathscr{A}$ is open, so $\mathscr{A}$ is also closed.