# What examples led to the modern definition of a topological space?

Today the language of topological spaces via open sets is fundamental in many different areas of mathematics, and it is a bit mysterious that the same formalism successfully captures such a wide variety of behavior. I can think of several independent reasons to invent the definition of a topology, all of which would have been on mathematicians' radar screens around the time the definition was first being mulled over in the early 20th century:

1. To provide a foundation for Klein's Erlangen program and Poincare's work on Betti numbers and the fundamental group
2. To clarify the foundations of calculus, e.g. the role of compactness in the extreme value theorem
3. To distinguish among various notions of convergence of functions (leading to functional analysis)
4. To give meaning to arguments involving "generic" configurations in algebraic geometry

My understanding is that it took quite some time for the modern formalism of topological spaces to emerge, so I'm wondering what specific results or examples were most influential in its development? And which modern applications of the theory were only realized after it matured?

• I think Volterra and some others (beginning in the mid or late 1880s, I believe) who started trying to make sense of calculus of variations methods by talking about doing calculus with "functions of curves" (e.g. their length), and Frechet's later unification of these ideas in his 1906 Ph.D. thesis, had much to do with the evolution of topology notions. See also the math Stackexchange question Origins of the modern definition of topology. Nov 12, 2014 at 18:56
• Its a good question as to why topology is introduced via open sets. When they were introduced to my physics class at college - they seemed distinctly unimpressed and the notion of open sets not at all natural to them. In fact, topology can be introduced via a generalisation of limits - which I expect to them would be far more natural. Liebniz already had the modern notion of continuity in embryonic form I believe. Sep 17, 2018 at 19:32

I believe that our modern definition of a topological space came primarily from Hausdorff's book Grundzüge der Mengenlehre (Foundations of Set Theory), first published in 1914, 2nd ed. 1927. Hausdorff started with metric spaces, but then generalized them.

Of course, the background to Hausdorff's work was the 19th work on continuity, and the so-called "arithmetization of analysis" --- the attempt to put calculus on a firm logical foundation. The biggest names here are Cauchy, Weierstrass, Dedekind, Bolzano, and Cantor. But the axiomatization of general topology in terms of open or closed sets is due to Hausdorff.

• The version I heard does, indeed, say it was Hausdorff. In the definition of a manifold, there are little neighborhoods mapped bijectively to open balls in Euclidean space, so that where they overlap the transition maps are continuous in Euclidean space. Then Hausdorff saw that in the definition of "continuous function" from $X \to Y$, you didn't need the neighborhoods to correspond to sets in Euclidean space, you could just say for each $a \in X$ and for each neighborhood $B$ or $f(a)$ in $Y$ there is a neighborhood $A$ of $a$ in $X$ such that $f$ maps $A$ into $B$. ... Sep 15, 2018 at 20:30
• ... So then he said: what if we take this as a definition of a type of space where we can define "continuous function". He gave axioms for this, where neighborhoods of points was the primitive notion. Later, others came up with other definitions, and Hausdorff's turned out to be a slight special case, and is now known as a "Hausdorff space". Sep 15, 2018 at 20:31
• @GeraldEdgar I heard the same story, with the twist that he was adapting the definition of a differential manifold to the more general continuous case. Also Weyl was supposed to be involved somehow. But I haven’t been able to track down where I read this. I couldn’t find it in Weyl’s The Concept of a Riemann Surface. Sep 16, 2018 at 21:14

Topological spaces appear to have their roots in the nineteenth century. It started, indirectly, with the theory of limits and delta-epsilon proofs. A major breakthrough occurred with the development of set theory (e.g. DeMorgan's Laws) in the middle to latter part of the century. This led to the "generalization" of limit, convergence, and accumulation point axioms using the theory of open and closed sets. Topology is sometimes referred to as "point set" theory.

The applications you cite came "later," that is, in the twentieth century. So did the so-called Separation Axioms, beginning with Hausdorff spaces, in 1914, and extended in the middle of the century. But the foundations for these applications were laid in the previous century.

• This does not answer the question at all, which specifically asks for examples of topological spaces. Your answer isn't completely unhelpful, but I think it would be better as a comment. Oct 31, 2014 at 10:50
• @JackM: In the question, the OP asked "what specific results or examples were most influential..." I answered using "results," not examples. You're a mathematician, and you deal in "examples." I'm a historian, and I deal in "timelines." (See our respective SE reputation scores.) From a historical perspective, "what led to" is well answered by results such as "limits and delta-epsilon proofs," as well as set theory. So my answer goes all the way back to the 19th century. For some people, that "big picture" may be as useful as contemporary examples. Oct 31, 2014 at 14:03