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Questions tagged [topology]

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

3
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2answers
91 views

Material on the History of Mathematical Spaces

First and foremost, I am aware that a similar question has been asked here and has been touched upon elsewhere. I have found these discussions very compelling but a bit light on external reference, ...
11
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2answers
172 views

History of various definitions of topology

I have been reading Point Set Topology for a while, and turns out that there are various possible ways to define a topology. Most popular one is using open set axioms. Another one is using closure ...
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3answers
98 views

Which book covers topology historically?

Is there a book, which expresses all the questions, or searches for attainment of certain utility/need, or thing, which gave the discovery or invention of all the components of Topology? I need book ...
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0answers
80 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
87 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
votes
1answer
116 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
183 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
70 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
186 views

Motivation of Continuous Functions

What is the historical motivation of continuous functions? Also, does anyone know who first isolated the idea?
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4answers
284 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 ...
8
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2answers
222 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
167 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
252 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 ...
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2answers
307 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
122 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
173 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
72 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
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1answer
109 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
249 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
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1answer
259 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
110 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
162 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
111 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
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2answers
125 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
134 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 ...
6
votes
1answer
288 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
220 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
115 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
278 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
128 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 .
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1answer
303 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
253 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
633 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?
15
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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?
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1answer
334 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 ...
12
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2answers
334 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 ...