Questions tagged [gn.general-topology]

Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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34 answers
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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 ...
230 votes
4 answers
15k 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 \...
Richard Dore's user avatar
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186 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(...
fedja's user avatar
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154 votes
4 answers
17k 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, $f(S)\subset Y$ ...
Willie Wong's user avatar
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136 votes
7 answers
32k 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 ...
Joseph O'Rourke's user avatar
133 votes
5 answers
29k 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 ...
Terry Tao's user avatar
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110 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 ...
Daniel Moskovich's user avatar
106 votes
9 answers
35k 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 ...
Kevin Buzzard's user avatar
103 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? ...
Brendan Guilfoyle's user avatar
90 votes
3 answers
13k 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 ...
Joel David Hamkins's user avatar
87 votes
19 answers
19k 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. ...
80 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. ...
Joel David Hamkins's user avatar
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 ...
Qiaochu Yuan's user avatar
76 votes
4 answers
14k 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”. ...
Claus's user avatar
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74 votes
9 answers
14k 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 ...
Aaron Mazel-Gee's user avatar
71 votes
3 answers
10k 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 ...
Andrea Ferretti's user avatar
71 votes
9 answers
8k 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 ...
Valerio Capraro's user avatar
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 ...
Ronald J. Zallman's user avatar
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
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 ...
Abcd's user avatar
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67 votes
11 answers
10k 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 $g: Y \to X$?
67 votes
3 answers
20k 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 ...
Martin David's user avatar
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65 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 ...
65 votes
14 answers
5k 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 ...
jon's user avatar
  • 801
62 votes
5 answers
7k 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 ...
JoeG's user avatar
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61 votes
7 answers
16k 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 ...
Anton Geraschenko's user avatar
58 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 ...
Ryan Budney's user avatar
  • 42.8k
57 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 ...
Gro-Tsen's user avatar
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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 ...
Andy Putman's user avatar
  • 43.4k
55 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 ...
Jake's user avatar
  • 795
55 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 ...
yada's user avatar
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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/...
Fedor Petrov's user avatar
53 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 $\mathbb{N}^2 \cong \mathbb{N}$, $\mathbb{R}^2\ncong \mathbb{R}, C^2\cong C$ (...
HenrikRüping's user avatar
53 votes
1 answer
4k 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 ...
Nick R's user avatar
  • 1,047
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. ...
50 votes
4 answers
22k 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 ...
Bugs Bunny's user avatar
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49 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 ...
Mark's user avatar
  • 4,804
49 votes
5 answers
8k 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 ...
Martin Brandenburg's user avatar
48 votes
3 answers
7k 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)...
Joseph O'Rourke's user avatar
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 ...
Pete L. Clark's user avatar
48 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 ...
57319's user avatar
  • 603
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 ...
Taras Banakh's user avatar
  • 40.8k
46 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 ...
André Henriques's user avatar
46 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 ...
Joel David Hamkins's user avatar
45 votes
2 answers
5k views

Continuous bijections vs. Homeomorphisms

This is motivated by an old question of Henno Brandsma. Two topological spaces $X$ and $Y$ are said to be bijectively related, if there exist continuous bijections $f:X \to Y$ and $g:Y \to X$. Let´s ...
Ramiro de la Vega's user avatar
44 votes
17 answers
16k 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....
44 votes
3 answers
12k 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 $\phi_*[N]\in H_*(M)$ ...
Steve's user avatar
  • 2,223
44 votes
6 answers
4k views

Does $\mathbb C\mathbb P^\infty$ have a group structure?

Does $\mathbb C\mathbb P^\infty$ have a (commutative) group structure? More specifically, is it homeomorphic to $FS^2$, (the connected component of) the free commutative group on $S^2$? $\mathbb C\...
Ben Wieland's user avatar
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