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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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Hahn-Banach theorem and ultrafilter lemma

I'm unable to understand a remark in "Two application of the method of construction by ultrapowers to analysis" by Luxemburg, which uses the ultrafilter lemma to prove the Hahn-Banach ...
oggius's user avatar
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4 votes
2 answers
292 views

$\mathrm{String}/\mathbb{CP}^{\infty}=\mathrm{Spin}$ or a correction to this quotient group relation

We know that there is a fiber sequence: $$ \dotsb \to B^3 \mathbb Z \to B \mathrm{String} \to B \mathrm{Spin} \to B^4 \mathbb Z \to \dotsb. $$ Is this fiber sequence induced from a short exact ...
zeta's user avatar
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145 views

Determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?

What and how to determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower? Namely, how do we know $$ K(Z_2,1)?, \quad K(Z_2,2)?, \quad K(Z,4)? $$ Naively -- in each step ...
zeta's user avatar
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0 answers
240 views

Examples of when $X$ is homotopy equivalent to $X\times X$

I was thinking about this question the other day: When is a topological space $X$ homotopy equivalent to $X\times X$ (with the product topology)? This is essentially a cross-post of this MSE question.....
pyridoxal_trigeminus's user avatar
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0 answers
165 views

Are all infinite-dimensional Lie groups noncompact?

Basically what the title says — if a Lie group is infinite-dimensional, is it necessarily noncompact?
Panopticon's user avatar
11 votes
1 answer
428 views

Is the Mandelbrot set Suslinian?

The Mandelbrot set is known to be (path-)connected and compact. A non-degenerate space with these properties is called a continuum. A continuum $X$ is Suslinian if every collection of non-degenerate ...
D.S. Lipham's user avatar
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2 votes
0 answers
92 views

Explicit CW-complex replacement of the space of reparametrization maps

Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-...
Philippe Gaucher's user avatar
10 votes
1 answer
392 views

Two dimensional perfect sets

Consider the following family of sets $$ \begin{align*} \mathcal{F} = \{X\subseteq [0,1]\times [0,1] \mid \ &X \text{ is closed and }\\& \forall x \in \pi_0 (X) (\{y \in [0,1] \mid (x,y) \in ...
Lorenzo's user avatar
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0 votes
2 answers
348 views

If a graph embedded on a surface is divided by a curve into a right and left that do not intersect can it be embedded on a surface of smaller genus?

Suppose we have a graph $G$ embedded on a (smooth, orientable etc) surface $Q$. Suppose there is a cycle $C$ of $G$ such that $C$ does not separate our surface $Q$ into two connected regions and ...
Hao S's user avatar
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1 answer
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Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynamical systems?

Because flowmaps are homeomorphic maps, I was wondering if there is any literature that proves that diffeomorphism $\Phi(x)$ can be expressed as a flowmap of a certain dynamical system? that is, does ...
li ang Duan's user avatar
1 vote
0 answers
141 views

Can a closed null-homotopic curve be filled in by a disc?

Let $U\subseteq\Bbb R^n$ be an open set and $\gamma\subset U$ a closed null-homotopic curve in $U$ (i.e. it can be contracted to a point). Then is there an embedded disc $D\subset U$ with boundary $\...
M. Winter's user avatar
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2 votes
0 answers
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Unordered configuration space with non-distinct points

Consider a topological space $X$, a natural number $n>0$ and the quotient topological space $Q_n(X)$ of $X^n$ by the equivalence relation : $x\sim y$ if and only if there is a permutation $\sigma$ ...
Phil-W's user avatar
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8 votes
1 answer
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Does the continuous image of a disc contain an embedded disc?

Let $\phi:\Bbb D^2\to\Bbb R^n$ be a continuous mapping of the 2-disc $\Bbb D^2$ that is injective on the boundary $\partial\Bbb D^2=\Bbb S^1$. Does its image contain an embedded disc with the same ...
M. Winter's user avatar
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3 votes
1 answer
161 views

Approximating continuous functions from $K\times L$ into $[0,1]$

Let $K$ and $L$ be compact Hausdorff spaces, let $f:K\times L\to [0,1]$ be continuous and let $\varepsilon>0$. Can we find continuous $g_{1},...,g_{n}:K\to[0,+\infty)$ and $h_{1},...,h_{n}:L\to[0,+\...
erz's user avatar
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11 votes
0 answers
172 views

Can the nowhere dense sets be more complicated than the meager sets?

Suppose $X$ is a completely metrizable space with no isolated points. Let $\mathcal{ND}_X$ denote the ideal of nowhere dense subsets of $X$, and let $\mathcal{M}_X$ denote the ideal of meager subsets ...
Will Brian's user avatar
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1 vote
1 answer
72 views

Is this notion of being "fully" convex closed under set addition?

While reading through "Linear Operators: General theory" by "Jacob T. Schwartz", reading the corollary to II.10.1 which states that for a compact convex subset $C$ of some ...
P. Quinton's user avatar
7 votes
0 answers
271 views

Generalizing uniform structures as Grothendieck topologies

Recently, I was reading a classical book "Sheaves in Geometry and Logic" by S. MacLane and I. Moerdijk, and then it stroke me that, that the definition of Grothendieck Topology bears some ...
Nik Bren's user avatar
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8 votes
0 answers
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A variation of necklace splitting

Our problem is the following: Let $n$ and $k$ be integers. We are given two (unclasped) necklaces, each with $n$ colored stones: a top necklace which has $k$ colors and a bottom necklace which has 2 ...
Sam King's user avatar
13 votes
2 answers
765 views

Smooth Urysohn's lemma on Fréchet spaces

Let $V$ be a Fréchet topological vector space. Let $K_0$ and $K_1$ be two closed subsets which are disjoint. I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$ whose restriction ...
André Henriques's user avatar
4 votes
0 answers
164 views

When $X$ is homeomorphic to $\mathscr{F}[X]$?

While I was talking to some colleagues, one of them said that there exists a topological space $X$ such that $X$ is uncountable, non-discrete and homeomorphic to $\mathscr{F}[X]$ (the Pixley-Roy ...
Carlos Jiménez's user avatar
8 votes
1 answer
351 views

"Compactness length" of Baire space

Intuitively, my question is: how many times do we have to mod out by an closed equivalence relation with all classes compact in order to collapse Baire space $\omega^\omega$ to a singleton? In more ...
Noah Schweber's user avatar
11 votes
1 answer
341 views

Density of linear subspaces in $C(K)$

Let $K$ be a compact Hausdorff space and denote by $C(K)$ the space of all real valued and continuous functions on $K$. We endow $C(K)$ with the supremum norm topology, making it a Banach space. ...
Julian Hölz's user avatar
7 votes
0 answers
150 views

The space of analytic associative operations

This question is a follow-up to this old one of mine. Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
Noah Schweber's user avatar
4 votes
3 answers
723 views

Does there exist a topological space $X$ such that $X^2$ and $[0,1]$ are homeomorphic?

I have proved that if $X$ is not connected then $X^2$ is not connected either. So my idea was to prove that if $X$ is connected then $X^2$ blown up any point is also connected. But I don't know ...
Fate Lie's user avatar
  • 505
2 votes
1 answer
300 views

If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected

Let $E\subset \mathbb{R}^n$ be a (measurable) subset with $\mathcal{H}^{n-1}(E)=0$, where $\mathcal H^{n - 1}$ is the ($n - 1$)-dimensional Hausdorff measure. I want to know if $\mathbb{R}^n\setminus ...
No-one's user avatar
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4 votes
0 answers
155 views

Two other variants of Arhangel'skii's Problem

This question is a follow up to another question of mine, which turned out to be easy (for background on Arhangel'skii's Problem see Arhangel'skii's problem revisited). Recall that a space is ...
Santi Spadaro's user avatar
2 votes
2 answers
274 views

Is a simple closed curve always a free boundary arc?

Is it possible to extract a neighborhood around any point on a simple closed curve such that the boundary of this neighborhood intersects the curve at only two points? For a simple closed curve $\...
S.Zhang's user avatar
  • 23
17 votes
3 answers
2k views

Is symmetric power of a manifold a manifold?

A Hausdorff, second-countable space $M$ is called a topological manifold if $M$ is locally Euclidean. Let $SP^n(M): = \left(M \times M \times \cdots \times M \right)/ \Sigma_m$, where product is done $...
Katrina's user avatar
  • 506
2 votes
0 answers
123 views

Homotopy type of a 3-manifold produced via Dehn surgery?

My apologizes if this is a fairly elementary question, I am still a novice when it comes to 3-manifold topology. I am wondering the following: by Kirby calculus, we know that two links (say in $S^{3}$ ...
Elliot's user avatar
  • 295
0 votes
2 answers
287 views

Distinguishable under manifold topology but indistinguishable under the Alexandrov topology

Take the time-oriented Lorentzian spacetime $(M, g)$ that is not strongly causal. In such case it is shown that the Alexandrov topology and the Manifolds topology deviate such that the manifold ...
Bastam Tajik's user avatar
8 votes
0 answers
192 views

Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?

I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question). Some simple ...
Pietro Majer's user avatar
  • 60.5k
2 votes
1 answer
264 views

Is a continuous functional on continuous functions the restriction of a continuous functional on the space of all functions?

As sets, we can consider the space $C(\mathbf{R}^n;\mathbf{R}^k)$ - of all continuous functions from $\mathbf{R}^n$ to $\mathbf{R}^k$ - to be a subset of the product space $(\mathbf{R}^k)^{\mathbf{R}^...
SBK's user avatar
  • 1,179
2 votes
1 answer
171 views

Is the collapse of a totally disconnected compact Hausdorff space still totally disconnected?

Let $S$ be a totally disconnected compact Hausdorff space and let $A\subset S$ be a closed subset. Let $S/A$ denote the space we get when collapsing $A$ to a point. Is this space still totally ...
user avatar
3 votes
0 answers
246 views

"weakly functorial resolution" of quasi-compact T_1 topological space by quasi-compact Hausdorff space

I have an arguably weird question: Let $X$ be a quasi-compact $T_1$ topological space, could there be a construction that takes such an $X$ as input and outputs a surjection $$X' \to X$$ with the ...
S. Li's user avatar
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5 votes
0 answers
249 views

Aspherical space whose fundamental group is subgroup of the Euclidean isometry group

Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...
Chicken feed's user avatar
0 votes
1 answer
109 views

Extending maps from a discrete set to a Stone-Čech compactification while retaining an injectivity condition

For $S$ a set, let $\beta_{\bf2}(S)$ be a compact, totally disconnected space containing $S$ where $S$ in the subspace topology is discrete and $S$ is a dense subspace, and $\beta_{\bf2}(S)$ has the ...
Tri's user avatar
  • 1,644
3 votes
2 answers
285 views

Cut a homotopy in two via a "frontier"

Consider a space $G$ obtained by glueing two disjoint cobordisms (the fact that they are might be irrelevant, assume they are topological spaces at first) $L$ and $R$ on a common boundary $C$. (...
Valentin Maestracci 's user avatar
5 votes
0 answers
131 views

Is the opposite of the category of $\kappa$-Lindelöf Hausdorff spaces locally presentable?

Gelfand duality tells us that the category of compact Hausdorff spaces (with continuous maps as morphisms) is contravariantly equivalent to the category of commutative, unital $C^\ast$-algebras (with $...
Tim Campion's user avatar
  • 63.9k
1 vote
1 answer
344 views

Is there anyway to formulate the Alexandrov topology algebraically?

One knows that the Alexandrov topology on a preordered set is the finest topology that induces the same [specialization] preorder on the set. Given this, one finds a one-to-one correspondence between ...
Bastam Tajik's user avatar
10 votes
0 answers
242 views

Arhangel'skii's problem revisited

One of the most well-known problems in set-theoretic topology is Arhangel'skii's question of whether there exists a Lindelöf Hausdorff space with "points $G_\delta$" (meaning, every point is ...
Santi Spadaro's user avatar
1 vote
0 answers
84 views

Is there a standard name for the following class of functions on non-Hausdorff manifolds?

Let $M$ be a (not necessarily Hausdorff) smooth manifold. Given an open chart $U\subset M$ and a compactly-supported smooth function $f:U\to\mathbb{R}$ on $U$, define $\widetilde{f}:M\to\mathbb{R}$ by ...
user49822's user avatar
  • 2,178
3 votes
1 answer
550 views

Do CGWH spaces form an exponential ideal in Condensed Sets?

If $X$ is any condensed set and $Y$ is a compactly generated weak Hausdorff (CGWH) space (a.k.a. $k$-Hausdorff $k$-space), is $Y^X$ again a CGWH space? To be more precise, is $(\:\underline{Y}\,)^X$ ...
user avatar
1 vote
0 answers
131 views

Can we construct a general counterexample to support the weak whitney embedding theorm?

The weak Whitney embedding theorem states that any continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold may be approximated by a smooth embedding provided $m > 2n$. ...
li ang Duan's user avatar
0 votes
0 answers
73 views

Can we construct general counterexample to support the Weak Whitney theorem? [duplicate]

Can we construct an example for the weak Whitney theorem to illustrate the existence of a continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold that cannot be smoothly ...
li ang Duan's user avatar
3 votes
0 answers
200 views

Contractibility of the pseudo-boundary of the Hilbert cube

Let the separable Hilbert cube $Q=\prod_{i=1}^{+\infty}[0,1]$ embed into the real Hilbert space $H=l^2(\mathbb{Z}^+)$, whose coordinate unit vectors are $\{ e_i \}_{i=1}^{+\infty}$, as the subset $\...
Zerox's user avatar
  • 1,543
0 votes
0 answers
161 views

Gluing faces of n-cube

Assuming $C_n$ be the $n$-cube, the intersection of $C_n$ with a supporting hyperplane $H(P, v)$ is called a face or more precisely a $d$-face if the dimension is $d$. Let $f_0$ and $f_1$ be faces ...
mahu's user avatar
  • 53
34 votes
6 answers
4k views

Why study finite topological spaces?

In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage: … this means that some concepts that I use freely and naturally in my personal thinking are foreign to ...
Wahome's user avatar
  • 737
4 votes
1 answer
179 views

A "simple" space with closed retracts but non-unique sequential limits

This question asking how KC ("Kompacts are Closed") and RC ("Retracts are Closed") are distinct has some good discussion, including a now-published example by Banakh and Stelmakh ...
Steven Clontz's user avatar
0 votes
0 answers
70 views

A cellular automaton with an image that is not closed

Let $G$ be a non-locally finite periodic group and let $V$ be an infinite-dimensional vector space over a field $\mathbb{F}$. Does there exist a nontrivial topology on $V^G$ and a linear cellular ...
mahdi meisami's user avatar
2 votes
1 answer
223 views

Is the projective limit $\mathcal{D}(\mathbb{R})$ separable?

Let $\mathscr{D}(\mathbb{R})$ be the set $C_0^\infty(\mathbb{R})$ of smooth functions with compact support endowed with the following topology: The initial topology with respect to the family maps $(\...
CoffeeArabica's user avatar

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