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Questions tagged [topology]

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

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Emmy Noether's announcement in 1932 ICM

I read a book "a history of abstract algebra"- chapter 6 by Israel Kleiner. And in this book, it is said that Emmy Noether gave a presentation at the ICM congress held in Zurich in 1932, ...
user1274233's user avatar
1 vote
0 answers
111 views

Relationship between electromagnetic and topological invariant

I read 17 equations that changed the world by Ian Stewart. This book provides information about the correlation between electromagnetic force and topological invariant. The idea of a topological ...
user1274233's user avatar
2 votes
0 answers
87 views

Origin of "Sierpinski space"?

Nowadays the unique 2 point, nondiscrete, nontrivial topological space goes by the name of the Sierpinski space. How did that space come to be named after Sierpinski? The comments to this MathOverflow ...
Lee Mosher's user avatar
1 vote
2 answers
223 views

Motivation and history of singular homology

Among the many cohomology theory's branches I asked about last time, I was curious about $d^2=0$ because I know that it is the formula that is the basis of all cohomology. So this time, I would like ...
user1274233's user avatar
3 votes
3 answers
259 views

History of cohomology theory

I saw this post. And I already posted it on Math stack exchange, but since someone recommended this site, I'm refining it and posting it again. And I understand that the mathematical object called ...
user1274233's user avatar
1 vote
0 answers
98 views

Was the "Gauss word realization problem" a kind of unknotting problem?

In Moritz Epple's article "Geometric Aspects in the Development of Knot Theory", Epple writes the following: It has been suggested that one of the earliest tools of combinatorial knot ...
user2554's user avatar
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67 views

Why the sphere eversion problem emerged?

Sphere eversion is the process of turning a sphere inside out in a three-dimensional space. See also this animation on YouTube: Outside in (2011) and picture below. My question is, what is the ...
Humberto José Bortolossi's user avatar
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0 answers
40 views

Lefschetz historical proof of Hyperplane Theorem

I would like to understand the the idea behind the historically original proof by Lefschetz of his Hyperplane theorem sketched roughly Here. The basic setup: Let $X$ be an $n$ -dimensional complex ...
user267839's user avatar
5 votes
0 answers
167 views

Early illustrations of topological notions in published work

Since I've not gotten any answers after a bit more than a week, I've now cross-posted to MathOverFlow. EDIT 2023-08-15: Several commenters here and at MO have asked me to sharpen the original question....
Sam Nead's user avatar
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4 votes
1 answer
185 views

How and when did the dedicated study of locally compact groups begin?

How and when did the dedicated study of locally compact groups begin? Specific instances from literature, recorded stories, etc., may help supplement the answers. There seems to be no reason why I ...
stoic-santiago's user avatar
0 votes
0 answers
109 views

Why is bachelors' unknotting called as such and who discovered it?

Bachelors' unknotting is a way to show that all tame knots are isotopic to the unknot, by tightening a knot to a point. Why is it called 'bachelors' unknotting'?
Apoorv Potnis's user avatar
4 votes
1 answer
201 views

Continuity, Hausdorff

Is the idea of a continuous map in the point-set model of topological spaces, i.e. that the preimages of opens are open, due to Hausdorff (Grundzüge der Mengenlehre)? For example, does the notion of ...
Hypatia du Bois-Marie's user avatar
3 votes
1 answer
135 views

Topologies without the axiom that finite intersection of open sets is open

A topology is a pair of a nonempty set $P$ of points, and a set $Opens\subseteq 2^P$ of open sets that is closed under two closure conditions: arbitrary (possibly infinite) unions and finite (...
Jim's user avatar
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0 answers
101 views

Why was it difficult to initially phrase compactness?

From beginnings of topology, it was clear that the closed interval $\left[a,b \right]$ of the real line had a certain property that was crucial for proving such theorems as the maximum value theorem ...
tryst with freedom's user avatar
4 votes
3 answers
894 views

The First Published book on Algebraic Topology

As far as I know, according to google, Eilenberg, Steenrod's book: Foundations of Algebraic Topology was published in 1952, and Spanier's book: Algebraic Topology was published in 1966. My Questions ...
Saikat's user avatar
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2 votes
0 answers
66 views

When was Lipschitz equivalence first attributed to Lipschitz or did Lipschitz formulate it himself?

In his book Introduction to Metric and Topological Spaces, author Wilson A Sutherland in explaining the equivalence of metrics invoked the definition: Two metrics $d_1, d_2$ on a set $X$ will be ...
User1865345's user avatar
5 votes
2 answers
269 views

Who is or was Stephen Barr, author of Experiments in Topology?

This concerns the Stephen Barr who wrote Experiments in Topology in 1964, available from Dover Publications. All I know about him is: He wrote the above book. He was a friend and possibly colleague ...
Prime Mover's user avatar
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3 votes
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What were the main different homology theories in 1930 and how did they get united?

I am reading The Uses and Abuses of the History of Topos Theory by Colin McLarty. On page 353, the following is said: An homology theory associates groups to topological spaces so that the group ...
tryst with freedom's user avatar
4 votes
0 answers
171 views

Why did the mathematical community settle on these properties to define a topology?

The following post is long, but I decided to write more rather than less in case it's helpful. I tried to make it clear, quick, and easy to skip to the short version of my question, so the reader can ...
Addem's user avatar
  • 511
-1 votes
1 answer
252 views

How long have all the mathematicians working in the respective fields known the theory of categories

Vague the question: how long have all the mathematicians working in the respective fields known the theory of categories? More specific questions: Is it true that all modern working algebraic ...
Arshak Aivazian's user avatar
1 vote
1 answer
196 views

Did the formation of Differential Geometry come before Differential Topology/ Topology in general?

I’m pretty interested in the history of mathematics, and it has always been my belief that the great pioneers of Differential Geometry were Gauss and Riemann, and the father of topology was mostly ...
Chris's user avatar
  • 111
2 votes
1 answer
182 views

Who proved Banach fixed point theorem in abstract metric spaces for the first time?

If one studies the paper written by Banach in which he first proved his fixed point theorem one would find that he did not prove the theorem for abstract complete metric spaces. He proved it for the ...
Aman Sharma's user avatar
6 votes
0 answers
158 views

How did Dyck originally state and prove his theorem in topology about the connected sum of a torus and projective plane?

Dyck's theorem in topology is sometimes stated as follows: the connected sum of a torus and projective plane is homeomorphic to the connected sum of three projective planes. Certainly, this is the ...
user avatar
1 vote
0 answers
100 views

Who extended the Banach fixed point theorem from the context of normed spaces to the context of metric spaces?

It is well known that Banach's fixed-point theorem was initially conceived as a fixed-point theorem for applications defined in normed spaces (see [1]). This theorem was conceived in 1922 by Stefan ...
MathOverview's user avatar
2 votes
2 answers
132 views

Who is the John Thomas of "Thomas's Plank" and "Thomas's Corkscrew"?

Entries $93$ and $94$ in Steen and Seebach's "Counterexamples in Topology" are named "Thomas's Plank" and "Thomas's Corkscrew". These are apparently named for a certain &...
Prime Mover's user avatar
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3 votes
3 answers
269 views

Historical ways of *presenting* or axiomatizing the notion of a topological space

I've heard rumors in a couple of places that the modern presentation of a topology as $(X, \tau)$ where $X$ is our topological space and $\tau$ is our set of open sets is "somewhat" new and ...
Greg Nisbet's user avatar
2 votes
0 answers
77 views

Development of the Idea of Homeomorphism?

How did the historical development of the notion of homeomorphism take place? At first, I think the most important thing to study were continuous functions and metric structures. When did the focus of ...
Saikat's user avatar
  • 348
6 votes
1 answer
219 views

What is the source of Hermann Weyl's remark about "near-sighted policemen" with respect to compact spaces?

Hermann Weyl is widely reported as making the following observation about a compact space: "If a city is compact, it can be guarded by a finite number of arbitrarily near-sighted policemen." ...
Prime Mover's user avatar
  • 1,289
5 votes
0 answers
93 views

Who first called the Brouwer Fixed Point Theorem "the crumpled paper theorem"?

Wikipedia attributes the remark to Brouwer himself, but I am extremely skeptical. Their citation goes to a webpage of a ? French educational TV show, where the remark appears to be a fictionalized ...
Nat Kuhn's user avatar
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3 votes
0 answers
104 views

Who invented the push-pull construction?

I learned about the push-pull construction from a video lecture by Freedman in which it is explained starting around 39:08. It is somewhat long and technical to describe in detail, but the main idea ...
Alessandro Codenotti's user avatar
1 vote
1 answer
425 views

Normed vector space : when and who?

When does the concept of "normed vector space" emerge? Who is the first mathematician to consider this setting?
Colas's user avatar
  • 113
7 votes
2 answers
900 views

On the history of Haar measure

Haar measure is a well-known concept in measure theory. Many books are perfectly dedicated to present its existence and uniqueness such as measure theory for D. Cohn. I am looking for a good ...
Neil hawking's user avatar
0 votes
0 answers
106 views

What if Newton's bucket had been a sphere?

My question is about Newton's bucket experiment. If a sphere filled (say) one-third with water is rotated very very fast, will the water eventually spread out across and coat the entire interior ...
Maxwell Cooper's user avatar
2 votes
2 answers
197 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, ...
LIAM M's user avatar
  • 21
16 votes
2 answers
2k 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 ...
Anirban's user avatar
  • 261
2 votes
3 answers
374 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 ...
Sensebe's user avatar
  • 675
1 vote
0 answers
169 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 ...
Hans-Peter Stricker's user avatar
2 votes
1 answer
410 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 ...
user2554's user avatar
  • 4,489
6 votes
1 answer
430 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 ...
user109871's user avatar
6 votes
2 answers
435 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 ...
User0112358's user avatar
3 votes
0 answers
110 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 ...
David's user avatar
  • 293
4 votes
1 answer
212 views

Motivation of Continuous Functions

What is the historical motivation of continuous functions? Also, does anyone know who first isolated the idea?
user109871's user avatar
1 vote
4 answers
2k 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 ...
José Hdz. Stgo.'s user avatar
9 votes
2 answers
335 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 ...
User0112358's user avatar
2 votes
0 answers
254 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 ...
Nick's user avatar
  • 121
5 votes
2 answers
404 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 ...
user2554's user avatar
  • 4,489
2 votes
2 answers
465 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 ...
Jesse P Francis's user avatar
2 votes
1 answer
220 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 ...
Catherine Ray's user avatar
6 votes
2 answers
335 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 ...
Pedro's user avatar
  • 163
0 votes
0 answers
112 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{...
David's user avatar
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