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, Ch.8.)
It could be shown that this definition is equivalent to the usual ones,
when restricted to connected open sets in the plane,
link.
(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).
Edit. As user46855 commented, it depends on
your definition of a “definition of simply connected''.
There are many equivalent definitions and, although I would be
very happy to receive some information about more recent ones
(that is, around and after 1900's), I am mostly interested in
the first definition, in whatever terms it was given.
At those times many terms might have been fluid, e.g. even though
I said above that Cauchy worked with simple closed curves (in 1814-1831),
Jordan's theorem only came at the end of that century.
So perhaps at those times there was no strict definition, but
at any rate someone must have stated something in specific enough terms,
so we would know they meant simply connected. User46855, thank you for the
J.C.Pont reference, will try to find it.
Edit March 20, 2017. The paper that motivated me to ask the above question was recently published in the American Mathematical Monthly. My coauthor and I adopted Poincare's work as reference to the (modern) definition of "simply connected", though we also indicate that some earlier work was done by Riemann
(and Gauss), and refer to Pont's book (given in a comment and an answer below). Thanks to user46855, Francois Ziegler, and to Moritz Firsching, for their comment's and/or answers. Our paper with Prof. J. Bak at CCNY is available at http://www.jstor.org/stable/10.4169/amer.math.monthly.124.3.217
Best Answer
It seems indeed pretty clearly to have been Riemann, in §6 of his Inaugural Dissertation (1851). There he defines zusammenhängend as well as einfach, zweifach and mehrfach zusammenhängend without citing any prior sources.
Yet a possible antecedent is Gauss, of whom J.-C. Pont's book (cited in the comments above and reviewed here) singles out a text, written around 1840, "which one can consider as a sketch of the theory of the order of connectivity". It would be interesting to dig Pont's exact reference, which ought to be available in Gauss's Werke.
(On Gauss and Riemann see also a letter of Betti published by A. Weil, Riemann, Betti, and the Birth of Topology, Archive for History of Exact Science 20 (1979) 91-96.)