Examples of arguments from connectedness Suppose $X$ is a connected topological space. A typical way that we prove a property $P(x)$ holds for all $x \in X$ is to show that $P$ is an open and a closed condition, and that $P(x)$ for some $x \in X$. 

Most recently I saw this show up in the proof that a connected, locally path connected space is path connected. (Pick a point $x \in X$ and let $P(y)$ be "There exists a path from $y$ to $x$".)

This seems to also be the idea behind some proofs that $[a,b]$ is compact. See here. 

I wanted to accrue some more examples of where we use this technique (for teaching purposes, perhaps). Does anyone have some more?
Uniqueness of lifting: Given a covering space $p \colon X' \to X$, a pointed function $f \colon (Y,y_0) \to (X,x_0)$ where $Y$ is connected and a value $x' \in p^{-1}(x_0)$, there exists at most one lift $f': Y \to X'$ such that $f = pf'$ and $f'(y_0)=x'$.

The proof uses the property $P(x)$: $f'_1(x)=f'_2(x)$; it is open, closed and true for $y_0$.
A nice one I saw recently in Warner's Foundations of Differentiable Manifolds and Lie Groups, in defining orientation on a manifold:

Suppose $M$ is connected. Consider $\bigwedge^{\text{top}}TM \setminus Z$, where $Z$ is the zero section of top-degree forms. Each fiber of $\bigwedge^{\text{top}}TM \setminus Z$ has exactly two components, so a connectedness argument (and local triviality of the bundle) shows that $\bigwedge^{\text{top}}TM \setminus Z$ must have at most two components. If it has one, the manifold is non-orientable; if it has two, $M$ is orientable.