Path-connectedness implies connectedness
= Path-connectedness implies connectedness
If $X$ had a separation into two disjoint nonempty open subsets, a <continuous path> joining points in the two subsets would pull it back to a separation of $[0,1]$. This contradicts <connectedness of an interval>. Thus every <path-connected space> is <connected>. The converse fails, for example for the <topologist's sine curve> with its limiting vertical segment.