
> 
> ***Q***. When did the notion of *homeomorphism* reach its 
>  modern formulation as a bicontinuous bijection, i.e., a 
>  continuous bijection
>  between topological spaces whose inverse is also continuous?
> 
> 
> 


Was this present in Riemann's work (1826-1866)?
Or in the work of Möbius (1790-1868); or Jordan (1838-1922)?
Was it clearly formulated by 1900?
Or only later, perhaps by Heegaard (1871-1948), or Dehn (1878-1952),
or Whitney (1907-1989)?


I am trying to understand the history behind the classification
theorem for compact surfaces, and it seems developing a precise notion of
"homeomorphism" was a key step.



Homeomorphism historically: When did it reach its modern formulation?
Well, all I did was a search on "homeomorphism history", but... I tried to extract some points that are made in conjunction to your question (Riemann, Möbius, Jordan), though feel free to edit it down if it is too long (and apologies to those who think this should be remapped to a History of Math Q/A).


**The evolution of the concept of homeomorphism**, by Gregory H. Moore  

Historia Mathematica, Volume 34, Issue 3, August 2007, Pages 333–343  

<http://www.sciencedirect.com/science/article/pii/S0315086006000863>


tl,dr: By 1930, the modern concept had been reached. From 1910-30 the definition (or common understanding of "topological invariance") moved definitively to bicontinuous (Fréchet, Hausdorff, and encyclopedists Zoretti and Rosenthal aiding this), whereas previously merely "continuous" was more popular (starting from von Dyck in 1888). Prior to that, the less-restrictive "deformations" (homotopies) were more prevalent in the thinking of Möbius, Jordan, and Klein (1860/70s), and while Poincaré (1895) coined the term homeomorphism, his meaning was that of diffeomorphism. Incidentally, Riemann had no notion of a topological mapping in his work.


Specifically regarding the question (unless otherwise indicated, all the below is from Moore's article):


**Section 2.** The evolution of the concept of “homeomorphism” was essentially complete by 1935 when Pavel Aleksandrov
(Paul Alexandroff) at the University of Moscow and Heinz Hopf at the Eidgenossische Technische Hochschule in
Zurich published their justly famous book *Topologie*, aiming to unify the two major branches of topology, the algebraic
and the set-theoretic. They took as their fundamental undefined concept “topological space,” based on the closure
axioms of Kazimierz Kuratowski [1922]. And they defined a homeomorphism between topological spaces in the
way that is now standard: “A one–one continuous mapping $f$ of a space $X$ into a space Y is called a topological
mapping or a homeomorphism (between $X$ and $f (X) = Y ⊆ Y$ ) if the inverse of $f$ is a continuous mapping of $Y$
to $X$. Two spaces... are called homeomorphic if they can each be mapped topologically onto each other” [Aleksandrov
and Hopf, 1935, 52].


Concerning the origins of topology, Aleksandrov and Hopf wrote: “We must regard Poincaré and Cantor as the
immediate founders of topology” [1935, 5]. So the reader might think that he should read the works of Poincaré and
Cantor if he wished to find the origin of the concept of homeomorphism. However, the reader would then find that
Cantor’s published works contain nothing at all about homeomorphisms, and very little about continuous functions, ...


[Quoting Johnson [1979, 127] about Riemann: What we find conspicuously lacking in Riemann’s work is the notion of a topological mapping. For modern mathematicians topology is inseparable from homeomorphisms. Riemann never contemplated these in his programme of analysis situs [i.e. topology].]


As for Poincaré, there is a particularly interesting discovery to be made. Although Poincaré coined the word
“homeomorphism” in [1895], he meant by it something quite different from and more restricted than what is meant
nowadays. ... his “homeomorphism” had strong requirements of differentiability and smoothness that have nothing directly to do with topology ... for Poincaré a “homeomorphism” is not a homeomorphism in the modern sense,
but rather a diffeomorphism. ... [Also noted is his 1892 article which spoke of deformations (homotopy), while an 1900 article of his might have used the word "homeomorphism" in the modern sense.]


**Section 3**. Since Poincaré’s 1895 article is the origin of the word “homeomorphism” but not of the concept of homeomorphism, how did that concept originate?
At the time that Poincaré wrote, there was a second and broader concept of homeomorphism in use. This second
concept was clearly stated by Walther von Dyck in an 1888 article on *analysis situs*: “Absolute properties [in *analysis
situs*] can also be characterized as those, for whose agreement on two manifolds, it is necessary and sufficient that
there exists a one–one continuous function [umkehrbar eindeutiger stetiger Beziehung] between all elements of the
two manifolds” [1888, 457]. [Footnote 4 notes that the German phrase could conceivably be interpreted either as "(umkehrbar eindeutiger) und stetiger" or "umkehrbar (eindeutiger und stetiger)", loosely "reversible mapping and continuous" and the question is whether "umkehrbar" (reversible) modifies "stetiger" (continuous); Moore notes that later mathematicians preferred the first construal.]


For the rest of the present note, a one–one continuous function from one subset of Euclidean space onto another
will be called a “$D$-homeomorphism,” after von Dyck, since a $D$-homeomorphism is a broader concept (both for
Euclidean spaces and more generally) than a homeomorphism. But more than three decades were to pass before
$D$-homeomorphisms were clearly distinguished from homeomorphisms. ...


It is a plausible conjecture that many of those who used $D$-homeomorphisms (something which continued to occur at least until 1924) implicitly assumed that every $D$-homeomorphism is a homeomorphism, i.e., that the inverse
function is continuous. Such a result was necessary, in particular, if the inverse of a $D$-homeomorphism was to be a
$D$-homeomorphism. And such a result would have been true if they had only considered, in $n$-dimensional Euclidean
space, sets which are both closed and bounded. But prior to Camille Jordan’s work [1893], discussed in Section 4
below, this condition was never made explicit. As we shall see when discussing Hurwitz [1898], there was a good deal
of murkiness around $D$-homeomorphisms. ...


In his article of 1863, Möbius wrote:



> 
> Two geometric figures are said to be *elementarily related* to each
>  other if each infinitesimal element (in all dimensions) of the one
>  figure corresponds to a similar element of the other in such a way
>  that if any two bounding elements of the one figure are contiguous,
>  then the same is true of the corresponding elements in the other; or,
>  what expresses the same thing, if a point of the one figure
>  corresponds to a point of the other, then any two infinitely close
>  points of the one figure correspond to infinitely close points of the
>  other. In this regard, a curve can only be elementarily related to a
>  curve, a surface only to a surface, and a solid body only to a solid
>  body. [1863, 435]
> 
> 
> 


Nothing in Möbius’s article settles the question of whether he was thinking of his “elementary relationships” as
homeomorphisms or as diffeomorphisms or as something different from both, such as deformations. All of his diagrams dealt with smooth two-dimensional manifolds. His use of infinitesimals suggests that he regarded his relations
as differentiable. (In any case, at the time he wrote, examples of continuous nowhere differentiable functions were
known only to a few mathematicians.)


The theorems stated in Möbius’s article would all remain true whether he was thinking of his elementary relationships as homeomorphisms, diffeomorphisms, or deformations. A typical such theorem was that a curve could be
elementarily related only to a curve, a surface to a surface, and a solid body to a solid body. Another theorem was that
two surfaces in a plane are elementarily related if and only if they have the same finite number of boundary curves
[1863, 439]. ...


[Later in the section, it is noted Möbius introduced the "rubber sheet" geometry (deformations) in this article/manuscript, similarly Jordan had deformations in his 1866 paper, and 10 years later Klein appeared to think in terms of deformations (which are homotopies, so less restrictive than homeomorphisms). Some comments are made about Schonflies (1906-8), Zoretti (1912), and Rosenthal (1924) the latter who finally distinguished between $D$-homeomorphism and homeomorphism: "For F. Klein [1872] and A. Hurwitz [1898] analysis situs [topology] is the study of those properties of figures (or point-sets) which are preserved by all one–one continuous mappings. It is altogether preferable to require here invariance with respect to one–one continuous mappings whose inverses are
also continuous. However, this distinction is only relevant for those figures which are not closed and bounded." Moreover, the article states "[t]here is evidence that even Felix Hausdorff, a decade before he introduced his version of topological spaces, shared
the erroneous view of Hurwitz [1898] and Schoenflies that topology is about the invariants of $D$-homeomorphisms."]


As late as 1924 some eminent algebraic topologists continued to use the older and inadequate definition of homeomorphism, i.e., of $D$-homeomorphism. Among those who did so was Solomon Lefschetz. ... However, six years later Lefschetz
adopted the modern definition of homeomorphism [1930, 3], and he retained this definition in later works [1942, 7]. In their 1934 textbook of algebraic topology, Herbert Seifert and William Threlfall (Dresden) wrote that “topology
has to do with those properties of geometric figures that are unaltered by topological mappings, i.e., one–one functions
such that they and their inverses are continuous” [1934, 1]. Thus Seifert and Threlfall accepted the modern definition
of homeomorphism. From about 1930, the modern definition has been dominant both in general topology and in
algebraic topology.


**Section 4**. (Homeomorphisms in abstract spaces)  

Already by 1906, spaces more abstract than $n$-dimensional Euclidean space had been proposed. In his doctoral
dissertation, Maurice Fréchet had been the first to introduce metric spaces (though under another name) as well as
his still more general L-spaces, which were based on an abstract notion of the limit of an infinite sequence of points
[1906]. In this context, Fréchet carried over from analysis the concept of a continuous function defined in terms of
sequences. ...


Although in 1906 Fréchet did not discuss homeomorphisms between two of his L-spaces, he did so four years later in an article in Mathematische Annalen on the concept of topological dimension. Given two sets $E\_1$ and $E\_2$, each part of an L-space, he wrote “that they are the image of each other, or that they are homeomorphic, if there exists between them a one–one correspondence which is bicontinuous [une correspondance biunivoque qui est bicontinue]”
[1910, 146]. And he was quite explicit that such a function was bicontinuous if and only if it and its inverse were continuous. This was the first time that the concept of homeomorphism was formulated in a more general context than
$n$-dimensional Euclidean space. [Footnote 6: Fréchet adopted the term “homeomorphism” from his teacher Hadamard, rather than directly from Poincaré.]


When Hausdorff formulated the concept of a Hausdorff topological space in his 1914 book, he generalized the
concept of a continuous function to such spaces. His definition was explicitly motivated by the epsilon–delta definition
of continuity of a real function due to Weierstrass. Since Hausdorff’s axioms for a topological space were in terms of
neighborhoods, his new generalized definition of continuous function was also formulated in terms of neighborhoods: ...


He then isolated the essential relationship between homeomorphisms and one–one continuous functions by giving
the condition under which a one–one continuous function is actually a homeomorphism, although he did so without
ever using the term homeomorphism: “If $B$ is the one–one continuous image of a set $A$ that is compact-in-itself [every infinite subset $A$ of $B$ has a limit point that belongs to $B$] then $A$ is also the continuous image of $B$." ...


It is surprising, under these circumstances, that nowhere in his 1914 book does Hausdorff define the concept of homeomorphism, and nowhere in it does he use the word “homeomorphism” or an equivalent term. Only in the second
edition of the book in 1927 did he define the concept of homeomorphism. There he gave the definition in a way that
suggests that he may have gotten it from Fréchet’s article of 1910. Hausdorff was thinking of a set $A$ and a set $B$,
together with a function $φ : A → B$ and its inverse function $ψ : B → A$ (he explicitly allowed both $φ$ and $ψ$ to be
many-valued). Then he wrote:



> 
> If, however, both functions $y = φ(x)$ and $x = ψ(y)$ are single-valued and 
>  continuous, then each of them will be called *reversibly continuous* or 
>  continuous from both sides or doubly continuous (fonction bicontinue). $B$ is 
>  also called a *homeomorphic image* of $A$.., and the one–one mapping between
>  both sets is called a *homeomorphism*. ... [1927, 195–196]
> 
> 
> 


The context in which Hausdorff gave this definition was that of metric spaces, since topological spaces were only discussed, very briefly, later in the book [1927, 226–232]. The French phrase “fonction bicontinue” that Hausdorff included in his original German text shows that he had
in mind some French author as the source of the underlying idea that a function and its inverse are both continuous.
Apparently this use of “bicontinue” originated with Fréchet [1910]. ...


Intriguingly, by 1921 the Polish school of topologists had ... surpassed French and German authors in understanding the difference between homeomorphisms and $D$-homeomorphisms, even in Euclidean spaces. [Kuratowski's 1921 solution of a 1920 problem posed by Sierpinski is noted: if $P$ is a 1-1 continuous image of $Q$ and vice-versa, must they be homeomorphic?] ...
Both Sierpinski and Kuratowski wrote important topology textbooks, Sierpinski in Polish [1928], which was then
translated into English [1934], and Kuratowski in French [1933]. In both of these books, homeomorphisms were
given the modern meaning. Thus by the early 1930s, homeomorphisms in the modern sense were the standard within
English, French, German, and Polish textbooks of topology.


