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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.

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I suspect that it depends from your definition of "definition". Dieudonne, history of algebraic and differential topology, says that before Poincare it was prehistory, and refers (for Betti, Riemann, and before) to J.C.Pont, la topologie algebrique des origines a Poincare, a book that unfortunately I cannot consult. –  user46855 Mar 22 at 20:54

2 Answers 2

up vote 14 down vote accepted

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.)

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I found J.C.Pont's book (but couldn't check it out or copy some of it). Gauss, Werke are also available here. Will work on this post again perhaps in a couple of days (planning to eventually accept the answers, but of course more answers would be welcome). –  user48481 Mirko Swirko Mar 24 at 21:59

The reference mentioned in Francois Ziegler's answer, that is given on page 37 in Pont's book is p.407-410 in volume 10.1 in Gauss's Werke:

"Bestimmung der Convergenz der Reihen, in welche die periodischen Functionen einer veränderlichen Größe entwickelt werden"

(It's a bit tricky to find the reference, since there is a typo where Pont gives the reference: 33 d should be exchanged with 33 c)

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Yes I had found p.37 in Pont's and suspected 33d was a typo, thank you for uncovering it. Also, p.71 in Pont discusses connexion simple and connection multiple. Google books here shows short paragraphs: keyword terms connexion Gauss and ordre de connexion produce results. Hathi Trust here allows very limited search (word count with page numbers). Will take a look again in Columbia Univ library (they won't let me check out books). –  user48481 Mirko Swirko Mar 24 at 12:30

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