Here is a question I saw today: Provide an example of a connected topological space having uncountably many path components.
Actually I have came up with an example which I believe should be correct:
Let $T$ be a subspace defined by $T=\{(x,\sin {1\over x}):0<x\le 1\}\cup \{(0,y):y\notin \mathbb Q, -1<y<1\}$. Since $T$ is a subset of the closure of a connected set $\{(x,\sin {1\over x}):0<x\le 1\}$, which $T$ contains, it follows that $T$ is connected. Also, each point $a\in \{(0,y):y\notin \mathbb Q,-1<y<1\}$ forms a path component $[a]$. It then follows that $T$ has uncountably many path components.
I found this example by noticing that the topologist's sine curve has two path components, one of which ($\{(0,y):-1\le y\le 1\}$) fills up the closure of the other component( $\{(x,\sin {1\over x}):0<x\le 1\}$.) When I take an uncountable collection of "separate points" in $\{(0,y):-1\le y\le 1\}$, each of which is a path component, I would yield a connected set with uncountably many path components. I am just wondering if one could find an example for this problem in a different manner from mine, because my approach feels a bit unnatural.