Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [topology]

Questions involving topology, the mathematical study of properties of spaces preserved by continuous maps.

6
votes
1answer
91 views

Origin of Compactness

According to Wikipedia https://en.wikipedia.org/wiki/Pavel_Urysohn, Urysohn and Alexandrov first formulated the modern definition of compactness. In which paper did they do this? Is there an English ...
5
votes
2answers
148 views

Who first described the fundamental group as the group of deck transformations?

Grothendieck developed the theory of the fundamental group of a scheme in SGA 1. In order to do so he used the fact that the fundamental group of a topological space is isomorphic to the group of deck ...
4
votes
0answers
66 views

History of Braids

I am looking for papers or books that describe the history of the development of braid theory, mainly during the 19th and the 20th century. I know Moritz Epple book on the history of the theory of ...
4
votes
1answer
170 views

Motivation of Continuous Functions

What is the historical motivation of continuous functions? Also, does anyone know who first isolated the idea?
1
vote
4answers
207 views

Who was the first individual that used the word “torus” to refer to $\mathbb{S}^{1} \times \mathbb{S}^{1}$?

Further, I believe that the idea to call it thus had to do with its resemblance to the "torus" in the base of some Greek columns of old: What do you think of this hypothesis of mine? Thanks in ...
7
votes
2answers
201 views

(Co)Homology: From topology to the rest of mathematics?

I can appreciate how (co)homology arose in the context of topology/geometry. Trying to get a handle on the handles of spaces leads one to this idea. It's not obvious, but I can see how this would lead ...
2
votes
0answers
158 views

Set Theory, onto and into their relation to spoken language definitions

Does anyone know how the definitions for onto and into map to the spoken language definitions of the words? I compared the Bourbaki definitions to these words and have a suspicion that the German ...
4
votes
2answers
220 views

What topological ideas did Gauss introduce to his student Möbius?

Recently I found a website with good historical information about the contributions of Gauss to Analysis situs (the old term for topology). The site is in German so I made a Google translate to ...
1
vote
2answers
224 views

Basic Theorems in Topology: Who proved them first?

Little thinking into basic Real Analysis results like Arbitrary union of open sets is open made me wonder who could have possibly proved it first - do we have any historic data on it? Also, who ...
1
vote
1answer
94 views

Who was Puppe of the Puppe sequence?

I have had difficulty locating the full name and story of the mathematician Puppe whose name adorns the beloved underlying long-exact sequence algebraic topology is built on. Does anyone know who they ...
6
votes
2answers
142 views

What exactly did Poincaré mean by 'simply connected'?

I've been reading John Stillwell's translation of the famous Analysis Situs and have become confused about the exact meaning of 'simply connected' in Poincaré's language. On page 7 (in the ...
0
votes
0answers
67 views

Riemann surfaces and covering

Assuming we have a Riemann surface $S$ of degree $n$ and we look at it as a covering of the projective line $\mathbb{P}^1$. If $B$ is the set of branch points of $S$ (when $B$ is a subset in $\mathbb{...
2
votes
1answer
93 views

Material models of Riemann surfaces

It is known that during the last quarter of the 19th century there was a flourishing of the production of material models (from plaster, strings, card-board etc) of curves and surfaces in Germany (but ...
10
votes
1answer
207 views

A knot cannot be tied in 4-dimensions, but when was this conjectured and proven?

Today it has been shown that a 1-dimensional object in 4-dimensional space cannot be tied into a knot. But I would like to know who first conjectured this and when? And when was it proven? (P.S., is ...
7
votes
1answer
224 views

Who first proved the “Cantor-Heine theorem” on uniform continuity?

The theorem is that any continuous function on a compact is uniformly continuous. It is called "Heine", and sometimes also "Heine-Cantor" theorem. My question is: what is the contribution of Cantor ...
4
votes
1answer
91 views

Origins and history of branched covering

During my research on branched coverings of the projective plane, I am interested to know the origins and history of branched coverings of the projective plane and the projective line, together with ...
0
votes
2answers
139 views

why was the hairy ball theorem important

In Topology courses one learns An even dimensional sphere does not possess any continuous field of unit vectors What is the importance of this result? I can't think of any applications off the top ...
2
votes
0answers
87 views

Why is the space of sections of $E$ called $\Gamma(E)$?

The space of sections of a bundle $\pi: E \to B$ is commonly denoted $\Gamma(E)$. (Note that the graph of a function $f$ is $\Gamma(f):=\left\{(x,f(x))\right\}$, and a particular section $\sigma: B \...
2
votes
2answers
116 views

Separability and second countability is the same thing to Halmos

I was browsing through Paul Halmos' classic book on measure theory from 1950, when I came by the following definition of separability on page 3 in the chapter on prerequisites: Today a separable ...
5
votes
0answers
123 views

Who discovered the singular cup product?

Cohomology is a stronger invariant than homology because it can be equipped with a ring structure. To be precise, if one starts with the singular cohomology groups $H^\bullet(-; R)$ with coefficients ...
7
votes
1answer
258 views

Who discovered the topological proof of Nielsen-Schreier theorem?

The celebrated Nielsen-Schreier theorem in group theory says subgroup of a free group is free. This was proved for finitely generated subgroups of free groups by Jakob Nielsen in 1921, which involved ...
3
votes
1answer
167 views

History of covering spaces

I want to know what lead to the notion of covering spaces, and the evolution of the concept. I understand that topology was not developed to solve problems, but to gain insight into the foundation of ...
2
votes
1answer
98 views

History of open cover compactness definition

I have been looking into the history of topology. One thing I am very curious about is the history of the open cover definition of compactness. According to Raman-Sundström, this goes back to a lemma ...
5
votes
1answer
244 views

Injection of Bernoulli numbers into topology

The Bernoulli numbers appear in the Harer-Zagier formula enumerating gluings of polygons, the Kervaire-Milnor formula for the order of homotopy groups for n-spheres, and (with the connection to the ...
-2
votes
1answer
120 views

questions about definition of topology [closed]

Do we have any information about definition of topology . Definition is not intuitive for me .Please share information about definition of topology .
9
votes
1answer
241 views

What caused the name change from “analysis situs” to “topology”?

J. Alexander's 1926 paper, Combinatorial Analysis Situs, doesn't refer to the field as combinatorial topology. He mentions that combinatorial analysis situs is concerned with topological invariants ...
5
votes
1answer
237 views

Did Gauss formulate, or at least know of, the full essence of the Gauss-Bonnet Theorem?

I know that a special case of the Bonnet theorem, called the Theorema Elegantissimum, was proved by Gauss in his 1827 treatise on differential geometry. This was a theroem that dealt with the ...
14
votes
1answer
553 views

How did Poincaré discover the fundamental group?

How did Poincare discover the fundamental group? What was the first instance that led Poincaré to discover this amazing theory?
13
votes
3answers
2k views

Who created topology, and when, and what problems lead to this creation?

Who and when created topology and how did it discovered the first time?
17
votes
1answer
311 views

What specific problems motivated Poincaré's work on topology?

The McTutor biography on Poincaré says: Poincaré's Analysis situs, published in 1895, is an early systematic treatment of topology. He can be said to have been the originator of algebraic topology ...
13
votes
2answers
313 views

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 ...