Walks, paths, connectedness and components in a simple graph on an arbitrary vertex set

Let $G=(V,E)$ be a simple graph in the sense of Simple graphs on an arbitrary vertex set.

A walk, closed walk, trail, path, and cycle in $G$ are defined exactly as in Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges, with no finiteness hypothesis on $V$. In particular, a walk is a finite vertex list $(v_0,\dots ,v_{\mathrm{\ell }})$ with $\{v_{i-1},v_i\}\in E$ for each $1\le i\le \mathrm{\ell }$, and a path is a walk with distinct vertices.

Vertices $u,v\in V$ are connected when some path in $G$ has endpoints $u$ and $v$. The graph $G$ is connected when $V\mathrm{\ne }\mathrm{\varnothing }$ and every two vertices of $G$ are connected.

For $v\in V$, the connected component of $v$ is the induced subgraph on $C_G(v):=\{u\in V:u\text{ is connected to }v\}\mathrm{.}$

These are the same notions as in Connected graphs and connected components defined by the existence of vertex paths, again with the finiteness hypothesis removed.