Path-connectedness and compactifications Is the compactification of a path-connected space path-connected? Why or why not? 
(I came across this question in my notes while studying for finals and I have no idea.) 

Let $\Gamma = \{(x, \sin(1/x)) \mid x>0 \}$ and $X=\Gamma \cup \{0\} \times [-1,1]$. Then $\Gamma$ is homeomorphic to $\mathbb{R}$ and $X$ is clearly a compactification of $\Gamma$. However, $X$ is not path connected.
