2
votes
1answer
29 views

Is there a name for spaces that always have local sections?

Given a continuous map $p:E \rightarrow B$ Suppose for every point $b \in B$ and a point $x \in p^{-1}b$ in the fibre of it, there is an open set $V$ of $B$ that contains the point $b$ such that ...
1
vote
0answers
28 views

Name of this property: all maps from given class of spaces into $X$ are nullhomotopic?

Let $X$ be a topological space and let $\mathcal{C}$ be some class of topological spaces. Is there a standard name for the following property of $X$? For every space $C\in \mathcal{C}$ all maps $C\to ...
1
vote
2answers
66 views

How to call two subsets that can be deformed into each other?

Given a topological space $X$, is there a canonical name for the equivalence relation generated by the following relation on the subsets of $X$? $A \sim B :\Leftrightarrow \exists \text{ continuous } ...
1
vote
1answer
28 views

$H^1(X) = [X,\mathbb{T}]$?

This is a stupid question, but here goes. I have a compact Hausdorff space $X$, and I am talking about $[X,\mathbb{T}]$, the group of homotopy classes of maps $X \to \mathbb{T}$, where $\mathbb{T}$ ...
3
votes
1answer
46 views

Local topological properties

Could we define the terms locally connected/compact/contractible/simply-connected/whatever to mean that there is a basis (for the topology on our space) of ...
7
votes
0answers
78 views

Origins of the name “Q” and “R” for cofibrant and fibrant replacement functors.

In a model category $\mathscr M$ (in the modern sense, i.e. closed and with functorial factorizations), there is a notion of fibrant and cofibrant replacement functors. Specifically, for any object ...
5
votes
1answer
129 views

What exactly is a dimension?

Maybe this is too broad a question, maybe I need to be more specific. I am just clearing my head here, feel free to ignore at your pleasure. In Linear Algebra, we learned that the dimension of a ...
2
votes
1answer
97 views

Confusion regarding various definitions in defining singular homology

In defining singular homology, A singular $n$-simplex is a continuous mapping $\sigma_n$ from the standard $n$-simplex $\Delta^n$ to a topological space $X$. Notationally, one writes ...
3
votes
0answers
177 views

Torsion-free fundamental group.

Is there a name for spaces whose fundamental group has no torsion? And what, if any, are some nice properties of these spaces?
7
votes
2answers
180 views

What is the gender of $K(\pi,n)$ in French?

This is a kind of silly question, but I don't know where else to ask. Suppose I wanted to say "Ceci n'est pas une pipe" but with $K(\pi,n)$ substituted for "pipe." Would the article be "un" or "une"? ...
16
votes
2answers
4k views

“A proof that algebraic topology can never have a non self-contradictory set of abelian groups” - Dr. Sheldon Cooper

In the current episode "The Big Bang Theory", Dr. Sheldon Cooper has a booklet titled "A proof that algebraic topology can never have a non self-contradictory set of abelian groups". I'm still an ...
1
vote
1answer
272 views

Meaning of commutative diagram

What is the meaning of a commutative diagram in mathematics? For example, if a map translate an object, then rotate it around the origin and then translate it again, is this a commutative ...
21
votes
1answer
529 views

An interesting topological space with $4$ elements

There is an interesting topological space $X$ with just four elements $\eta,\eta',x,x'$ whose nontrivial open subsets are $\{\eta\},\{\eta'\},\{\eta,\eta'\}, \{\eta,x,\eta'\}, \{\eta,x',\eta'\}$. This ...
1
vote
0answers
126 views

Terminology in an Exercise of Hatcher

I am trying to solve an exercise in Hatcher "Algebraic Topology" but am a little confused by the terminology he is using (just so you know, it is exercise 5 in chapter 1.1). He that in the problem ...
3
votes
0answers
67 views

Is there are a name for a simplicial complex that is homotopic to the clique complex of its 1-skeleton?

A hollow octahedron is a nice triangulation of the sphere, because once you know the edges, you know everything. The vertices are obviously the ends of the edges, and the faces are any collection of ...
2
votes
1answer
158 views

Extension and reduction of the structure group

Let $H\subset G$ be a subgroup and $\pi:P\to B$ be a principal $H$-bundle. $G$ has a left $H$ action and one can define a principal $G$-bundle $\pi':P\times_H G\to B$ where $P\times_H G$ is ...
2
votes
1answer
88 views

Shellable and Graphs

Suppose we have a graph $G$ of order $n$. Also suppose that we form the coloring complex $S(G)$ of $G$. What does it mean when we say that $S(G)$ is shellable?
21
votes
3answers
1k views

Why algebraic topology is also called combinatorial topology?

I remember reading somewhere(at least more than once) that algebraic topology is also known by the name "Combinatorial Topology" which essentially tags the subject fundamentally with some counting ...
2
votes
1answer
263 views

Meaning of “topologically equivalent”

I've been wondering about "topologically equivalent" for some time now. For example: $S^1$ is "topologically equivalent" to $\mathbb{R}P^1$. I see that they are homotopy equivalent. But are they ...
14
votes
1answer
535 views

When does variété mean manifold?

Following advice from this post, I am in the process of translating Ehresmann's 1934 paper "Sur la Topologie de Certains Espaces Homogènes" from French to English. French-English dictionaries online ...
8
votes
2answers
254 views

Etymology of the name “deck transformation”

What does the word "deck" mean in "deck transformation"? What's the idea behind this name?
14
votes
3answers
2k views

What is a natural isomorphism?

I came across the adjective "natural" many times in my reading and I think this has something to do with category theory. Could someone please illustrate the idea behind this adjective to me? Many ...