The following definition is due to Donald J. Newman: 
A connected open subset $D$ of the plane $\mathbb C$ 
is _simply connected_ 
if and only if its complement $\widetilde D = \mathbb C \setminus D$ 
is ``connected within $\varepsilon$ to $\infty$'', 
that is, if for any $z_0 \in \widetilde D$ and $\varepsilon > 0$ 
there is a continuous curve $\gamma(t)$, $0 ≤ t < \infty$ such that:

(a) dist$(\gamma(t), \widetilde D) < \varepsilon$ for all $t \ge 0$,

(b) $\gamma(0) = z_0$,

(c) $\lim\limits_{t\to\infty} \gamma(t) = \infty$. 

see the textbook _Complex Analysis_, by Joseph Bak and Donald J. Newman. 
(His definition is easy to use, allowing _induction_ on the number of 
levels of a polygonal path, towards the proof of the Closed Curve Theorem.) 

It could be shown that this definition is equivalent to the usual ones, 
when restricted to connected open sets in the plane, 
[link](http://at.yorku.ca/cgi-bin/abstract/cbgb-57). 
(But not in $\mathbb R^3$, where $D=\mathbb R^3$ minus the $z$-axis is a counterexample.) 

Question 1. Who first defined _simply connected_ ? 

Cauchy did a lot of work related to independence of 
an integral on the path of integration 
[see e.g. the book _Cauchy and the Creation of Complex Function Theory_, 
Frank Smithies, Cambridge Univ.Press, 1997], 
but it appears that he usually worked with simple closed curves, 
and paths inside their bounded component, without ever explicitly using the term, 
or the notion, of a _simply connected domain_. 

On the other hand in the book [_History of Topology_, ed. I.M. James, 
North Holland, 2006] several authors discuss _simple connectedness_. 
In particular R. Vanden Eynde, p.73 (again, related to independence 
on the path of integration) indicates that Riemann (as in 1851) 
distinguished between simply connected and multiply connected surfaces, 
citing: 
_This leads to a distinction between simply connected surfaces, in which any closed
curve is the boundary of a part of the surface - like for instance a circle - and multiply
connected surfaces, for which this property is not valid, - like for instance the surface
bounded by two concentric circles_. 

Question 2. Was it Riemann who _first_ defined _simply connected_? 
Did Cauchy ever consider this notion explicitly? Did anyone between 
Cauchy and Riemann consider it (or even earlier, though this seems unlikely), 
any references? 

Nowadays there are several equivalent definitions, they appear to have come later, 
which might perhaps be the subject of another discussion. 

Any help is greatly appreciated (either on the questions stated above, or any 
comments on various equivalent definitions known).