Questions tagged [topology]
Questions involving topology, the mathematical study of properties of spaces preserved by continuous maps.
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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 ...
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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 ...
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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....
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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 ...
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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'?
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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 ...
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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 (...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 &...
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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 ...
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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 ...
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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."
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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 ...
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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 ...
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Normed vector space : when and who?
When does the concept of "normed vector space" emerge?
Who is the first mathematician to consider this setting?
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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 ...
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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 ...
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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, ...
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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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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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(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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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{...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \...
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