Optional: Explain why the “topologist’s sine wave” \[X=\{(0,0)\}\cup\{(x,\sin(1/x)):x>0\}\] is an example of a space which is connected but not path connected.
Solution
Let \[ G=\{(x,\sin(1/x)):x>0\}. \] The map \[ (0,\infty)\to G, \qquad x\mapsto(x,\sin(1/x)) \] is a homeomorphism, with inverse given by first-coordinate projection. Hence \(G\) is path connected, in particular connected.
The point \((0,0)\) lies in \(\overline G\): for example, \[ x_n=\frac1{n\pi}\longrightarrow0, \qquad (x_n,\sin(1/x_n))=(x_n,0)\longrightarrow(0,0). \] A connected set together with any one of its closure points is connected, so \[ X=G\cup\{(0,0)\} \] is connected.
It is not path connected. Suppose there were a path \[ \gamma(t)=(x(t),y(t)):[0,1]\to X \] from \((0,0)\) to a point of \(G\). Let \[ t_0=\sup\{t:x(t)=0\}. \] Then \(x(t_0)=0\), while for \(t>t_0\) sufficiently close to \(t_0\) we have \(x(t)>0\) and hence \[ y(t)=\sin(1/x(t)). \] Continuity of \(x\) and \(x(t_0)=0\) force \(1/x(t)\) to become arbitrarily large as \(t\downarrow t_0\). Since \(x([t_0,t_0+\varepsilon])\) is an interval containing \(0\) and some positive number, it contains values \(1/(\pi/2+2\pi n)\) and \(1/(3\pi/2+2\pi n)\) for arbitrarily large \(n\). Along corresponding sequences \(t_n,s_n\downarrow t_0\), one has \[ y(t_n)=1, \qquad y(s_n)=-1, \] contradicting continuity of \(y\) at \(t_0\). Thus no such path exists, and \(X\) is connected but not path connected.