Connected space infinite path components

Question

Simply put are there connected spaces with infinitely many path components?

Answer 1

Yes. Take, in $\mathbb R^2$,$$\{0\}\times\mathbb R\cup\bigcup_{n\in\mathbb Z}\left\{\left(x,2n+\sin\left(\frac1x\right)\right)\,\middle|\,x>0\right\}.$$It is connected and it has countably many path components: every $\left\{\left(x,2n+\sin\left(\frac1x\right)\right)\,\middle|\,x>0\right\}$ is such a component.

Answer 2

As a more extreme example, take the so-called pseudo-arc https://en.wikipedia.org/wiki/Pseudo-arc (the simplest nondegenerate hereditarily indecomposable continuum, not that is sounds simple). It is connected, of cardinality continuum, and each path-component is a singleton.