All Questions
5,184 questions
333
votes
34
answers
96k
views
Why is a topology made up of 'open' sets? [closed]
I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...
231
votes
4
answers
16k
views
Is $\mathbb R^3$ the square of some topological space?
The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X \...
- 5,275
184
votes
8
answers
12k
views
Two commuting mappings in the disk
Suppose that $f$ and $g$ are two commuting continuous mappings from the closed unit disk (or, if you prefer, the closed unit ball in $R^n$) to itself. Does there always exist a point $x$ such that $f(...
- 61.9k
155
votes
4
answers
18k
views
Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not?
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
- 39.1k
140
votes
7
answers
34k
views
Is the boundary $\partial S$ analogous to a derivative?
Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after ...
- 151k
135
votes
5
answers
31k
views
Does the inverse function theorem hold for everywhere differentiable maps?
(This question was posed to me by a colleague; I was unable to answer it, so am posing it here instead.)
Let $f: {\bf R}^n \to {\bf R}^n$ be an everywhere differentiable map, and suppose that at each ...
- 114k
113
votes
4
answers
13k
views
Is there a sheaf theoretical characterization of a differentiable manifold?
I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...
- 22.1k
107
votes
9
answers
36k
views
solving $f(f(x))=g(x)$
This question is of course inspired by the question How to solve f(f(x))=cosx
and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...
- 41.4k
105
votes
5
answers
16k
views
Independent evidence for the classification of topological 4-manifolds?
Is there any evidence for the classification of topological 4-manifolds, aside from Freedman's 1982 paper "The topology of four-dimensional manifolds", Journal of Differential Geometry 17(3) 357–453? ...
- 2,337
92
votes
3
answers
14k
views
Is every sigma-algebra the Borel algebra of a topology?
This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, Correspondence
between Borel algebras and topology.
Since the question was not answered there after some ...
- 236k
91
votes
19
answers
20k
views
Injectivity implies surjectivity
In some circumstances, an injective (one-to-one) map is automatically surjective (onto). For example,
Set theory
An injective map between two finite sets with the same cardinality is surjective.
...
82
votes
5
answers
6k
views
How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?
This question is about the space of all topologies on a
fixed set X. We may order the topologies by refinement, so
that τ ≤ σ just in case every τ open set is open in σ.
...
- 236k
79
votes
5
answers
5k
views
Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem?
The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective ...
- 118k
78
votes
12
answers
12k
views
Why aren't representations of monoids studied so much?
It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit ...
- 2,392
77
votes
4
answers
15k
views
What are good mathematical models for spider webs?
Sometimes I see spider webs in very complex surroundings, like in the middle of twigs in a tree or in a bush. I keep thinking “if you understand the spider web, you understand the space around it”. ...
- 6,937
76
votes
9
answers
15k
views
understanding Steenrod squares
There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from ...
- 6,069
75
votes
3
answers
11k
views
Cohomology and fundamental classes
Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...
- 14.7k
72
votes
9
answers
9k
views
What is a continuous path?
I would like some help, because I am getting mad trying to answer the following
Question: Let $X$ be a topological space, what is a continuous path in $X$?
Well, maybe you're already getting ...
- 6,034
71
votes
1
answer
2k
views
Dualizing the notion of topological space
$\require{AMScd}$
Defining a topological space on a set $X$ is equivalent to designating certain subobjects of $X$ in ${\bf Set}$ (monomorphisms into $X$ up to equivalence) as open. The requirements ...
user30211
70
votes
28
answers
7k
views
Examples where it's useful to know that a mathematical object belongs to some family of objects
For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon:
(1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
68
votes
3
answers
21k
views
Properly Discontinuous Action
When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex ...
- 1,236
68
votes
2
answers
2k
views
Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$
Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a ...
- 629
67
votes
22
answers
10k
views
When has discrete understanding preceded continuous?
From my limited perspective, it appears that the understanding
of a mathematical phenomenon has usually been achieved,
historically, in a continuous setting
before it was fully explored in a discrete ...
67
votes
11
answers
11k
views
How should one think about non-Hausdorff topologies?
In most basic courses on general topology, one studies mainly Hausdorff spaces and finds that they fit quite well with our geometric intuition and generally, things work "as they should" (sequences/...
67
votes
10
answers
12k
views
Non-homeomorphic spaces that have continuous bijections between them
What are nice examples of topological spaces $X$ and $Y$ such that $X$ and $Y$ are not homeomorphic but there do exist continuous bijections $f: X \to Y$ and $g: Y \to X$?
66
votes
5
answers
8k
views
Does homology have a coproduct?
Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor ...
- 661
66
votes
4
answers
6k
views
Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
Similarly is the complement of any countable set in $\mathbb R^3$ simply connected?
Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...
- 1,187
65
votes
14
answers
6k
views
Notions of convergence not corresponding to topologies
This question concerns the ramifications of the following interesting problem that
appeared on Ed Nelson's final exam on Functional Analysis some years ago:
Exam question: Is there a metric on the ...
- 801
61
votes
1
answer
5k
views
Every real function has a dense set on which its restriction is continuous
The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous.
Or so I'm told, but this leaves me ...
- 32.5k
60
votes
7
answers
17k
views
Is there a measure zero set which isn't meagre?
A subset of ℝ is meagre if it is a countable union of nowhere dense subsets (a set is nowhere dense if every open interval contains an open subinterval that misses the set).
Any countable set ...
- 24.1k
60
votes
6
answers
7k
views
Torsion in homology or fundamental group of subsets of Euclidean 3-space
Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups is not ...
- 44.4k
58
votes
8
answers
9k
views
Is there a Whitney Embedding Theorem for non-smooth manifolds?
For smooth $n$-manifolds, we know that they can always be embedded in $\mathbb R^{2n}$ via a differentiable map. However, is there any corresponding theorem for the topological category? (i.e. Can ...
- 825
56
votes
17
answers
13k
views
Atiyah-Singer index theorem
Every year or so I make an attempt to "really" learn the Atiyah-Singer index theorem. I always find that I give up because my analysis background is too weak -- most of the sources spend a lot of ...
- 44.8k
56
votes
3
answers
3k
views
Duality between compactness and Hausdorffness
Consider a non-empty set $X$ and its complete lattice of topologies
(see also this thread).
The discrete topology is Hausdorff. Every topology that is finer than a Hausdorff topology is also ...
- 1,773
54
votes
4
answers
6k
views
Are the rationals homeomorphic to any power of the rationals?
I asked myself, which spaces have the property that $X^2$ is homeomorphic to $X$. I started to look at some examples like $\mathbb{N}^2 \cong \mathbb{N}$, $\mathbb{R}^2\ncong \mathbb{R}, C^2\cong C$ (...
- 11.1k
54
votes
3
answers
9k
views
If any open set is a countable union of balls, does it imply separability?
If a metric space is separable, then any open set is a countable union of balls. Is the converse statement true?
UPDATE1. It is a duplicate of the question here
https://math.stackexchange.com/...
- 109k
53
votes
4
answers
24k
views
When is $L^2(X)$ separable?
I have never studied any measure theory, so apologise in advance, if my question is easy:
Let $X$ be a measure space. How can I decide whether $L^2(X)$ is separable?
In reality, I am interested in ...
- 12.3k
53
votes
3
answers
8k
views
Grothendieck's manuscript on topology
Edit: Infos on the current state by Lieven Le Bruyn: http://www.neverendingbooks.org/grothendiecks-gribouillis
Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (...
52
votes
7
answers
8k
views
"Algebraic" topologies like the Zariski topology?
The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact.
...
51
votes
5
answers
9k
views
Fundamental group as topological group
Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...
- 63.1k
51
votes
3
answers
3k
views
Is each squared finite group trivial?
A semigroup $S$ is defined to be squared if there exists a subset $A\subseteq S$ such that the function $A\times A\to S$, $(x,y)\mapsto xy$, is bijective.
Problem: Is each squared finite group ...
- 41.9k
50
votes
7
answers
5k
views
Is there an algebraic approach to metric spaces?
It is well known that most topological spaces can be studied via their algebra of continuous real-valued (or complex-valued) functions. For instance, in the setting of compact Hausdorff spaces, there ...
- 4,874
49
votes
3
answers
8k
views
Thurston's 24 questions: All settled?
Thurston's 1982 article on three-dimensional manifolds1 ends with $24$ "open questions":
$\cdots$
Two naive questions from an outsider:
(1) Have all $24$ now been resolved?
(2)...
- 151k
48
votes
19
answers
17k
views
What is your favorite proof of Tychonoff's Theorem?
Here is mine. It's taken from page 11 of "An Introduction To Abstract Harmonic Analysis", 1953, by Loomis:
https://archive.org/details/introductiontoab031610mbp
https://ia800309.us.archive....
48
votes
3
answers
13k
views
When is a Homology Class Represented by a Submanifold? [duplicate]
Possible Duplicate:
Cohomology and fundamental classes
Given an oriented manifold $M$ and an oriented submanifold $\phi:N\to M$ we can obtain a homology class $\phi_*[N]\in H_*(M)$ ...
- 2,283
48
votes
6
answers
4k
views
Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?
In his 1967 paper A convenient category of topological spaces,
Norman Steenrod introduced the category CGH of compactly generated Hausdorff spaces
as a good replacement of the category Top topological ...
- 43.2k
48
votes
8
answers
8k
views
When are there enough projective sheaves on a space X?
This question is being asked on behalf of a colleague of mine.
Let $X$ be a topological space. It is well known that the abelian category of sheaves on $X$ has enough injectives: that is, every ...
- 65.4k
47
votes
4
answers
4k
views
Which topological spaces admit a nonstandard metric?
My question is about the concept of nonstandard metric space that would arise from a use of the nonstandard reals R* in place of the usual R-valued metric.
That is, let us define that a topological ...
- 236k
47
votes
3
answers
3k
views
A metric characterization of the real line
Is the following metric characterization of the real line true (and known)?
A nonempty complete metric space $(X,d)$ is isometric to the real line if and only if for every $c\in X$ and positive real ...
- 41.9k
47
votes
4
answers
3k
views
Is Schauder's conjecture resolved?
Schauder's conjecture: "Every continuous function, from a nonempty
compact and convex set in a (Hausdorff) topological vector space into
itself, has a fixed point." [Problem 54 in The ...
- 593