Questions tagged [topology]
Questions involving topology, the mathematical study of properties of spaces preserved by continuous maps.
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0answers
80 views
Has Harnack's proof of parity of countable intervals ever been defended?
in MSE we find that Axel Harnack in 1885 proved that the interval [0,1] can be covered by a countable number of small intervals such that a countable number of intervals remains in the complement. ...
1
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0answers
71 views
Were the ancient Greeks aware of the “topology” of (Euclidean) space?
Related to a more mathematically inclined question, I'd like to ask the following question:
The ancient Greeks made use of infinite arguments and processes (limits), e.g. in the method of exhaustion ...
2
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0answers
79 views
Several questions about Gauss's mathematical conception of braids
I'm trying to figure out several things about Gauss's thoughts concerning a certain four-strand braid. The reference my questions are based on is mainly Moritz Epple's excellent article "orbits of ...
6
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1answer
102 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
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2answers
162 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
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0answers
69 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
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1answer
185 views
Motivation of Continuous Functions
What is the historical motivation of continuous functions? Also, does anyone know who first isolated the idea?
1
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4answers
236 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
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2answers
214 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
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0answers
163 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
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2answers
241 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
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2answers
264 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
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1answer
110 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
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2answers
160 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
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0answers
69 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
105 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
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1answer
245 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
247 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 ...
5
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1answer
103 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
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2answers
155 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
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0answers
100 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
121 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
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0answers
129 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
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1answer
272 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
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1answer
186 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
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1answer
105 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
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1answer
265 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 ...
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1answer
124 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
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1answer
275 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
245 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
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1answer
593 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?
14
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3answers
3k 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
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1answer
319 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
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2answers
325 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 ...
