A domain $D$ in $C$(complex number) in which every holomorphic function of $g\in O(D)$ is integrable is called "homologically simply connected."
$\textbf{Q:}$ How does the definition imply $D$ is connected or path-connected to start with? Normally, $O(D)$ being domain implies $D$ connectedness. However, the statement seems to imply that the dual space(i.e. integration along cycles) will pick out non-connectedness?
Ref: Remmert Complex Analysis Chpt 9. Miscellany, Sec 3.2
In fact, Remmert denotes an open subset as a domain and a connected open subset as a region. Therefore "homologically simply connected" does not imply "connected".
Remmert also states that homologically simply connected domains are precisely the topologically simply connected ones. This is wrong if we use the standard definition of a simply connected space (which includes that it is pathwise connected). We can only say that each component of a homologically simply connected domain is simply connected.
However, Remmert does not give a definition of "topologically simply connected". Perhaps he understands it in the sense that each loop is contractible. This would be highly unusual, but seems to be the only explanation for his statement.