In Hatcher's discussion on covering spaces, he mentions "If $p: \tilde{X} \rightarrow X$ is a covering space, then the cardinality of the set $p^{-1}(x)$ is locally constant over $X$. Hence if $X$ is connected, this cardinality is constant as $x$ ranges over all of $X$. It is called the number of sheets of the covering.''
It is true that if $\mathscr P$ is a local property (i.e., true in arbitrarily small neighborhoods, i.e., given any point $x\in X$ and any open neighborhood $U$, there is a neighborhood $V$ of $x$ such that $V\subseteq U$ and $\mathscr P$ holds in $V$) of a connected space $X$, then $\mathscr P$ is true globally? Is there a precise version of this statement?
Edit: It has been pointed out that my guess was not correct. Still, there are examples such that this holds. For instance, a connected & locally path connected space is path connected. Lukas also pointed out that any locally constant function on a connected space is constant. Are there more examples like these?
No. The circle $S^1$ is connected and locally simply connected, but not globally simply connected.