Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,432
questions
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Complement of contractible locally Euclidean subspace
Let $X$ be a connected closed topological manifold. Let $S\subset X$ be a contractible locally Euclidean subspace. Is $X\setminus S$ connected?
1
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0
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152
views
Homotopy groups of ball complement
Let $X$ be a connected closed topological manifold. Let $n$ be an integer such that $\pi_i(X)=\{0\}$ for $1\leq i \leq n$.
Let $f:B^m\to X$ be a topological embedding, where $B^m$ is the $m$-...
4
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0
answers
266
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Sierpinski's characterization of $F_{\sigma\delta}$ spaces
According to [2]: Let $X$ be a space. We call a system $(X_s)_{s\in T}$ a Sierpinski
stratification of $X$ if $T$ is a nonempty tree over a countable alphabet and $X_s$ is a closed subset of $X$ for ...
4
votes
1
answer
162
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Consider a net of weak order units in a Riesz space converging in order to a weak order unit. Is there a tail whose infimum is a weak order unit?
Let $X$ be an extremally disconnected (the closure of an open set is open) compact Hausdorff space, and consider the Riesz space $C^\infty(X)$ of continuous functions from $X$ to the extended real ...
2
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1
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170
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Set of null-sequences is not $\sigma$-compact
I am interested in a reference for the following fact (or a similar result).
PROPOSITION. Let $X$ denote the set of real null sequences; i.e., the set of $(a_n)_{n=0}^{\infty}$ with $a_n\to 0$, with ...
6
votes
2
answers
986
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Foundational results dependent on/equivalent to the continuum hypothesis or its negation?
I remember at a certain point early in my mathematical studies learning that the Axiom of Choice is equivalent to the following statement on Cartesian products:
If $\{ X_i \}_{i \in I}$ is any ...
16
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2
answers
3k
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Is there a conceptual reason why topological spaces have quotient structures while metric spaces don't?
Of the mathematical objects that I am familiar with, it is normally the case that the product of 2 objects is an object of the same type and that an equivalence relation on an object induces a ...
18
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3
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Is a inverse limit of compact spaces again compact ?
Then one can construct a model for the inverse limit by taking all the compatible sequences.
This is a subspace of a product of compact spaces. This product is compact by Tychonoff. If all the spaces ...
1
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1
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148
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Regularity and ultrafilter
I read the following result in an article.
Let $X$ be a regular space. Let $\mathcal{M}$ be free closed
ultrafilter on $X$. Set $\mathcal{U=}\left\{ U:U\text{ is open and there
exists a }F\in \mathcal{...
0
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1
answer
169
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P-filter property?
Let $\mathcal{F}$ be a $P$-filter on $\omega$. Denote by $\Omega=\bigsqcup \omega_i$ where $\omega_i=\omega$. Consider the $P$-filter $\mathcal{S}$ on $\Omega$ whose base is as follows
$(\bigsqcup_i ...
6
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2
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175
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Nonhomeomophic spaces with homeomorphic mapping cones
It is natural to ask if it is possible for the mapping cone $X\cup_\alpha CA$
to be homeomorphic to the mapping cone $X\cup_\beta CB$ with $A$ and $B$
nonhomeomorphic. Is there a standard go-to ...
7
votes
1
answer
411
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Proper homotopy
Let $F: X \times [0, 1] \to Y$ be a homotopy such that for any $t \in [0,1]$ the map $F( \cdot, t) : X \to Y$ is proper. Is it true in general that $F$ is proper?
I am interested in particular in ...
1
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1
answer
270
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Comparison of product topology and colimit topology in sequence spaces
In Munkres Theorem 20.4 it is shown that the (relative) uniform topology induced by:
$$
d(x,y)\triangleq \sup_{n \in \mathbb{N}} d(x_n,y_n)
$$
is strictly finer than the product topology on $\prod_{n \...
1
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1
answer
198
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Is every homeomorphism approximately a product of homeomorphisms?
Let $\phi$ be a homeomorphism on $\mathbb{R}^{n+m}$, $\epsilon>0$, and $K\subseteq \mathbb{R}^n$ be a non-empty compact. Does there necessarily exist homeomorphisms $\phi_1,\phi_2$ on $\mathbb{R}^...
2
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1
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230
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Approximate selection for finite-valued upper hemicontinuous/semicontinuous maps?
I'd like to know if there are any known-results on the existence of continuous approximation theorems for upper hemicontinuous (aka upper semicontinuous) maps $\phi: X\rightarrow Y$ which are finite ...
5
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2
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157
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Local cross-sections for free actions of finite groups
Let $G$ be a finite group, let $X$ be a locally compact Hausdorff space, and let $G$ act freely on $X$. It is well-known that the canonical quotient map $\pi\colon X\to X/G$ onto the orbit space $X/G$ ...
11
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2
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497
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Continuous version of the fundamental theorem of invariant theory for the orthogonal group
A standard result in the invariant theory of the orthogonal group states the following.
Theorem
Let $(E, \langle .,. \rangle)$ be an n-dimensional euclidean vector space,
let $f : E^m \rightarrow {\bf ...
0
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0
answers
160
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Problem of Thickening an Arc in a Topological $ 2 $-Manifold
Let $ M $ be a topological $ 2 $-manifold (possibly with boundary), $ C $ an arc in the interior of $ M $ (i.e., an injective continuous function from $ [- 1,1] $ into $ \operatorname{Int}(M) $), and $...
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83
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Compact metrizable contractible locally contractible topological space of finite covering dimension is a CW complex
Let $X$ be a compact metrizable contractible locally contractible topological space of finite covering dimension. Is $X$ homeomorphic to a CW complex?
5
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1
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445
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Any continuous map is homotopic to one assuming fixed values at finitely many points
Let $X$ and $Y$ be topological spaces. Assume $X$ is locally contractible and has no dense finite subset. Assume $Y$ is path-connected.
Given $n$ pairs of points $(x_i, y_i)$ where $x_i\in X$ and $y_i\...
1
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1
answer
70
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Projecting Graph of a Function acted on by a homeomorphism
Let $X,Y$ be compact, connected, simply-connected, and separable, metric spaces each with at-least $2$-points, and let $f,g:X\rightarrow Y$ be continuous functions. Does there always exist a ...
2
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1
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103
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When almost all points are not isolated in all subspaces
Let $X$ be a compact (non Hausdorff) $T_0$ topological space such that for any subset $\mathcal{A}=\{\mathfrak{x}_\alpha\}_{\alpha\in \Lambda}$
of distinct element of $X$ the set $\{\mathfrak{x}_\beta\...
7
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6
answers
2k
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How to partition R^3 into pairwise non-parallel lines?
Problem. How to partition R^3 into pairwise non-parallel lines?
A possible solution is to stack infinitely many ``concentric'' hyperboloids, by increasing radius and decreasing slope. And don't forget ...
11
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1
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446
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How much of general topology can be developed by taking the notion of "connected set" as the sole topological primitive
Let X be an infinite regular topological space which is connected and locally connected.
Question. If no point of X is a cut point, does X always have base of connected open sets whose complements (...
7
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0
answers
216
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adding one point from the Stone-Cech compactification
Let $X$ be any non-compact Tychonoff space and $\beta X$ be its Stone-Čech compactification.
The following fact is known: any point $p$ from the reminder $\beta X \setminus X$ is not a $G_{\delta}$-...
2
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0
answers
70
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Separating a certain planar region with an open set
I have a fairly specific question related to plane separation properties. I couldn't quite see how to use Phragmen–Brouwer properties to answer it because those kind of results generally apply to ...
4
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0
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344
views
When every closed and connected subset is path connected
Let $X$ be a compact $T_0$ topological space such that its closed and connected subsets are path connected. Is there any characterization for such a space?
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2
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209
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Intrinsically defining smooth/continuous/analytic functions
In mathematics, the notion of a continuous/smooth/analytic function $\mathbb{R}\to\mathbb{R}$ is introduced by defining the general set-theoretic function $\mathbb{R}\to\mathbb{R}$ and then imposing ...
2
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1
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186
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Are there minimal topological conditions on a space 𝑋 for it to have a countable separating set?
Are there minimal topological conditions on a space $X$ for it to have a countable separating set?
A separating set here is a set $D \subset C(X)$ (where $C(X)$ is the space of continuous functions ...
4
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0
answers
125
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'Monodromy' for relative homology group
Let $A$ and $X$ be topological manifolds. Denote by $\mathbb {Emb}(A,X)$ the space of all topological embeddings $A\to X$.
A loop $f_s:A\to X$ ($s\in[0,1]$) in $\mathbb{Emb}(A,X)$ should give rise to ...
2
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1
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68
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Equicontinuity-like property of a convex compact set
Let $X$ be a Tychonoff topological space and let $x\in X$. Let $B\subset C(X)$ be convex and compact in the topology of pointwise convergence, and such that $f(x)=1$, for every $f\in B$.
Is there an ...
5
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1
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338
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Mapping $\mathbb P$ onto $\mathbb Q ^\omega$
Let $\mathbb P$ denote the space of irrationals. Is there a continuous bijection (one-to-one and onto) $f:\mathbb P\to \mathbb Q ^\omega$ that maps each closed subset of $\mathbb P$ to a $G_\delta$-...
4
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1
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296
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Functoriality of Atiyah-Hirzebruch spectral sequence - Reference Request
I'm interested in a text book reference on the functoriality of the Atiyah–Hirzebruch spectral sequence. The only reference I found are these lecture notes by Kupers (link should lead to the target ...
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45
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Critical Growth of Dimension for Dense Cover by Linear Subspaces
Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that
For any sequence of distinct finite-dimensional ...
5
votes
1
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204
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Arc connectedness of product spaces
Is arc connected-ness well-behaved with respect to products?
That is -
$\prod X_\alpha$ is arc connected iff $X_\alpha$ is arc connected $\forall \alpha$
In this question on MathStackexchange, an ...
4
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1
answer
229
views
Measurable total order
Under what conditions on a metric space $X$, equipped with the Borel $\sigma$-algebra, does there exist a measurable total ordering of the elements of $X$?
By "measurable total ordering" we ...
32
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1
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2k
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Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?
On page 205 of his Topology textbook, James Munkres made an interesting remark:
It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer ...
3
votes
2
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307
views
A minimal continuum
A continuum $X$ is called minimal if it is not a single point and is homeomorphic to all its nontrivial subcontinua.
Here a trivial continuum is a single point.
What is an example of a minimal ...
3
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0
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72
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A holomorphic shrinking of a domain into a compact subset
This question is related to these two.
Let $X\subset \mathbb{C}^{n}$ be a bounded domain. I am interested in the following property: there is a sequence of continuous maps $\varphi_n:\overline{X}\to X$...
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1
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312
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Basis for space of continuous, surjective monotone functions on $\mathbb{R}$ [closed]
$\DeclareMathOperator\CM{CM}$
I recently came across Okhezin - Study of families of monotone continuous functions on Tychonoff spaces describing monotone functions on general topological spaces and I ...
5
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0
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144
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Do products of distance functions separate points?
Let $(X,d)$ be a metric space without isolated points and of diameter $1$. Let $Y=\{y_m\}_{m=1}^{\infty}$ be a dense subset of $X$.
Define $g_0\equiv 1$, and for $m>0$ let $g_m=d(\cdot,y_1)\dotsm d(...
1
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0
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107
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Is the Vietoris topology on compact subsets of $\mathbb R^n$ locally convex?
The title question says it all really.
If the question is negative for compact subsets of $\mathbb R^n$, is it affirmative for compact and convex subsets of $\mathbb R^n$? How about for all nonempty ...
9
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250
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Does a generalization of Tietze's extension theorem hold for set-valued functions?
Let $X$ be a normal topological space. Tietze's extension theorem says that if $A \subset X$ is closed, then a continuous function $f: A \to \mathbb R^n$ can be extended to a continuous function whose ...
3
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0
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67
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Is there a complex Katetov-Tong theorem?
For any bounded function $f: X \to \mathbb R$, not-necessarilly continuous, one can define for any $x$, the real functions
$$
\limsup f(x) = \inf_{U\in\mathcal V_x} \sup_{u\in U} f(u)
$$
and
$$
\...
1
vote
1
answer
753
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Known dense subset of Schwartz-like space and $C_c^{\infty}$?
After reading this question, which asked for some examples of commonly used (proper) dense subsets of $C_0^{\infty}(\mathbb{R}^n)$ with the $L^p$-norm I wonder. What are some "well-known" ...
0
votes
1
answer
75
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Ultrabornological representation for the space of uniformly continuous functions?
Let $\{\omega_i\}_{i\in I}$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $0$. Then, for each $i \in I$ define the space
$$
C_{\omega_i}(\mathbb{R}^n,\...
3
votes
1
answer
139
views
Is each preseparable topological group narrow?
A topological group $G$ is defined to be
$\bullet$ precompact if for any neighborhood $U\subseteq G$ of the unit there exists a finite subset $F\subseteq G$ such that $G=UF$;
$\bullet$ narrow if for ...
2
votes
1
answer
102
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Isomorphic hypergraphs of dense sets of Hausdorff spaces
If $H_i = (V_i, E_i)$ are hypergraphs for $i = 1,2$ , we say that they are isomorphic if there is a bijection $f:V_1 \to V_2$ such that for all $e\subseteq V_1$ we have $e\in E_1$ if and only if $f(e)\...
1
vote
2
answers
315
views
Connectedness of compact metric space
Let $X$ be a compact metric space satisfying the following condition: for any given positive number $\delta>0$, only finitely many components of $X$ have diameter larger than $\delta$.
For a given ...
1
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0
answers
63
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Connected components of bounded linear operators of $V = (\mathcal C(U(1), \mathbb C) , \lVert \cdot \rVert_\infty)$
This question is related to this one.
Consider the complex Banach space $V=(\mathcal C(U(1), \mathbb C), \Vert \cdot \Vert_\infty)$ where $\mathcal C(U(1), \mathbb C)$ is the space of continuous ...