- What are the connected components and path connected components of $\Bbb{Q}$ , $\Bbb{R}$ and $\Bbb{R}_\mathcal{l}$?
Definition: A component $C$ of a topological space $X$ is a maximal connected subspace.
I think every connected component of $\Bbb{R}_\mathcal{l}$ consists of just one point. Because, if we take any subset say $S$ with at least two distinct points of $\Bbb{R}$ then $S$ can't be connected. If $x,y\in \Bbb{R}$ be two distinct points, such that $x<y$ then $A=(-\infty,x)\cap S$ and $B=(y,\infty)\cap S$ are disjoint non-empty open sets in the subspace topology on $S$. Therefore, $S$ is disconnected. If the above proof is true for $\Bbb{R}_\mathcal{l}$.
Similar proof for path component.
Is this even true for $\Bbb{R}$? I mean $\Bbb{R}$ has components that consists of just one point.
I believe $\Bbb{Q}$ has also connected components that consists of just one point.
Any help or hint will be appreciable. Thanks!
You are right about $\Bbb R_\ell$ and $\Bbb Q$. For instance, $\{(a,b):a,b\in\Bbb R\setminus\Bbb Q\text{ and }a<b\}$ is a clopen base for $\Bbb Q$, so $\Bbb Q$ is totally disconnected (and even zero-dimensional). $\Bbb R$, however, is path-connected and hence also connected: for any $a,b\in\Bbb R$, the map
$$f:[0,1]\to\Bbb R:x\mapsto a+(b-a)x$$
is a path from $a$ to $b$.