All Questions
5,184 questions
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Subset of the domain of attraction
Let $x \in R^n$ and $f : R^n \to R^n$, $f\in C^1$
$$
\frac{\mathrm{d}}{\mathrm{d}t} x(t) = f(x(t))
$$
be such that $f(0) = 0$ is asymptotically stable. The domain of attraction is the set of initial ...
- 61
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225
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Is it possible to prove that surfaces with compact boundary are homeomorphic by glueing disks to the boundary components?
Let $S_1$ and $S_2$ be two surfaces with compact boundary and the same number of boundary components. Let $M_1$ and $M_2$ be the surfaces obtained by glueing closed disks to the boundary of $S_1$ and $...
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0
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155
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$f:Y\to X$ continuous with $f^{-1}(x)$ compact for $x\in X$, does there exist a Borel measurable map $g:X\to Y$?
Let $X,Y$ be Polish, metric spaces. $f:Y\to X$ is a continuous, surjective map and for any $x\in X$, $f^{-1}(x)\subset Y$ is compact. Is it true that there is a injective, Borel measurable map $g:X \...
1
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0
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78
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Words with finite critical exponent
Let $\mathcal{A}$ be a finite set. Is there a nice characterization of the subset of $S\subset \mathcal{A}^\omega$ such that every $w\in S$ has finite critical exponent? Of course $S$ has measure zero ...
- 4,459
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289
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About Whitehead's problem
Hi I am new to proofs of consistency and independence with ZFC of some claims. I have read "The uses of set theory" by Judith Roitman, in that article it is mentioned that the Whitehead ...
- 561
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0
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282
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Regular measure in a Hausdorff space
Let $(X, \beta, \mu)$ be a measure space, and $(X, \tau)$ be a Hausdorff topological space such that:
$\mathcal{B}(\tau)\subset\beta$; where $\mathcal{B} (\tau)$ is the Borel set generated by $\tau$.
...
- 111
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83
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What is known about the algebraic completion of a monoid?
It is the monoid obtained by adjoining all solutions of polynomial equations. I'll demonstrate how to adjoin a single solution to a polynomial equation to a monoid:
Let $W$ be a monoid and let $p(x)=q(...
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231
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Has an "algebraic manifold" been defined before? Are there any non-trivial examples?
Let $S$ be a set and $\cdot$ a partial binary operation on $S$. A subset $F\subseteq S$ is $\cdot$-closed if the following condition holds:
for all $f,g\in F$, if $(f,g)\in\mathrm{dom}(\cdot)$, then $...
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102
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What is the real name for the initial object in the category of "monoid-valued measures of intervals" on transitive relations?
(I'm not asking for a true/false answer; I have a true algebraic fact and I'm looking for a reference in the literature. By the way, there is a version of this theorem that replaces monoid with $R$-...
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52
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Is every cyclic right action of a cancellative invertible-free monoid on a set isomorphic to the set of shifts of some homography?
The terms are defined in a related question. [1]
Conjecture 1. Let $A$ be a set, $W$ a cancellative invertible-free monoid, and $\cdot\colon A\times W\rightarrow A$ a cyclic right $W$-action generated ...
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0
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71
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terminology for a kind of two-sided module over a monoid
If $M$ is a monoid object in a pointed category $\mathcal{C}$, then a right $M$-module is an object $X$ equipped with a morphism $\alpha: X\times M\to X$ that satisfies the usual rules. There are ...
- 12.5k
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57
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Sequence in local cohomology for multiple closed subsets
Let $X$ be topological space with closed subsets $A,B,C \subset X$ and $\mathcal{F} \in Sh(X)$.
I'm trying to understand
\begin{equation*}
H^i_{A\cap B}(X,\mathcal{F}) \oplus H^i_{A\cap C}(X,\mathcal{...
- 473
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81
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Examples of spaces which have explicit expression as colimits in $\mathrm{Top}$
$\DeclareMathOperator\Ball{Ball}$Question: What "well-known" spaces can be explicitly written down in the form $\bigcup_k \phi_k C(K_n,\mathbb{R}^m)$; where $K_n$ is a non-empty compact ...
- 5,405
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355
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On logarithmic schemes
I have two questions on logarithmic schemes
Can we explicitly construct a chart for any coherent logarithmic scheme? By definition of coherence it must have a chart but given a coherent sheaf of ...
- 494
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2k
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Weak sequential continuity vs strong continuity
Let $E$ be a Banach space, $T:E\rightarrow E$ a non-linear operator.
$T$ is said to be Weakly Sequentially Continuous (shortly W.S.C) on $E,$ if for every $\left(x_{n}\right)_{n}\subseteq E$ with $x_{...
- 291
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824
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A weakly sequentially continuous operator which is not weakly continuous
I'm reading some papers where the condition of weak sequential continuity is crucial instead of the weak continuity.
So, let
$T$ an operator between a Banach space $X$ and itself.
$T$ is weakly ...
- 291
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52
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A local base for space of probability measures with Prohorov metric
Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...
- 111
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152
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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?
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154
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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 $m$-...
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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?
user145520
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120
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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 ...
- 869
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65
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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 ...
- 1,355
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59
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About Countable Dense Homogeneous spaces (CDH) and strongly locally homogeneous spaces
I am new to the study of CDH topological spaces, I wanted to study basic examples of this type of spaces, for example I could understand the demonstration that $\mathbb{R}$ is CDH, using the Cantor ...
- 561
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0
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79
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Conditions for a function to vanish almost nowhere on its support?
Let $f:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous function and $\mathrm{supp}(f) := \mathrm{cl}\{x\in\mathbb{R}^d\mid f(x)\neq 0\}$ its support.
Under which conditions is it true that $f≠0$ (...
- 463
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470
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Cellular chain complex of $G$-CW-complexes & their differentials
I not completly understand EXAMPLE 2.31 (page 19) dealing with homology of
$G$-CW-complexes. Source: http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/LTCC-notes-Lecture3-2019.pdf
...
- 6,038
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379
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Creating an inverse system which "stratifies density"
Setting:
Let $X'$ be a dense subset of an infinite-dimensional Fréchet space $X$ and suppose that $(X_n')_{n \in \mathbb{N}}$ is a nested sequence of non-empty subsets of $X'$ satisfying
$$
\bigcup_{n ...
- 5,405
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102
views
Functions preserving almost disjoint of partitions
A collection $\mathcal{A}\subseteq \omega^\omega$
is almost disjoint iff
$\bigcap_{X\in \mathcal{A}}X^{-1}(j)$ is finite for all $j\in\omega$.
A function $\Gamma:2^\omega\rightarrow 2^\omega$ is
...
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43
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Continuous surjection from $X(D_n)$ onto $\operatorname{Homeo}_0(D_n)$
Let $n>1$ and let $\mathfrak{X}(D_n)$ denote the set of continuous vector fields on the closed disc $D_n\subseteq \mathbb{R}^n$. Let $\operatorname{Homeo}_0(D_n)$ be the set of homeomorphism of ...
- 5,405
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53
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Spaces that are comparable with their compacts
This is an outgrowth of this question.
For a (metrizable) space $X$ consider the following increasingly strong properties:
(i) For every compact $K\subset X$ there is a map $f:X\to X$ such that $K\...
- 5,529
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104
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"Global" topologies between compact convergence and uniform convergence
Let $X$ and $Y$ be locally compact (but not compact), second countable, Hausdorff spaces with $Y$ metric. It is easy to see that the topology of compact convergence is weaker than the topology of ...
- 5,405
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52
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Is a finite increasing chain of closed sets the closure of the union of the interiors of the relative complements? [closed]
Let $X$ be a topological space. Suppose there are closed subsets $X=:F_{k}\supseteq F_{k-1}\supseteq\cdots\supseteq F_{1}\supseteq F_{0}:=\emptyset$. Is it true that
$\overline{\bigcup_{j=1}^{k}\...
- 267
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61
views
Minimal radius of a ball admitting a trivialization of a vector bundle
Let $X$ be a compact Hausdorff space and $p : V \to X$ a complex vector bundle of rank $n$. For $r > 0$ let $B(r,x)$ denote the open ball of radius $r$ around $x$. Does there exist an $r$ such that,...
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0
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27
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Approximation of multipliers by multipliers of a smaller set 2
This question is a refinement of my previous question.
Let $X$ be a compact metric space, and let $B$ be a bounded Banach Disk in $C(X)$ such that for every $x\in X$ there is $f\in B$ with $f(x)\ne 0$...
- 5,529
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224
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Surjectivity of colimit maps for topological spaces
From this post and to (co)completness of the category Top of topological spaces and continuous functions we know that for any diagram $B_i$ and an object $A$ in Top, there are natural maps of sets:$\...
- 5,405
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142
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Identifying the two points of a subspace homeomorphic to a Sierpinski space
Let $X$ be a $\Delta$-generated space having a subset $A=\{a,b\}$ such that the relative topology is the Sierpinski topology with for example $\{a\}$ closed and $\{b\}$ open (the Sierpinsky space is a ...
- 3,901
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135
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A density problem
Let $\langle\cdot,\cdot\rangle$ be the usual scalar product in ${\bf R}^n$ ($n\geq 2$) and let $B$ be the closed unit ball of ${\bf R}^n$.
Denote by $C^0(B,B)$ the space of all continuous functions ...
- 293
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216
views
Uniform convergence over compacts subsets implies existence of a uiform convergente subsequence?
Let $H$ the group of all homeomorphisms of a locally compact second countable and totally bounded metric space $X$ onto itself, under the compact-open topology ($X$ is totally bounded if every ...
- 365
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117
views
Stone duality- a modification
Let $2$ be the discrete topological space with two elements. For a map of sets
$$\beta : X \times Y \rightarrow 2 $$
We get a topology on $X$ and a topology on $Y$. The topology on $X$ is the weakest ...
user30211
1
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30
views
The cellularity of the composition of cellular maps (with arbitrary CW decompositioning)
Is the composition of cellular maps cellular?
Related to this, I have another question. (I apologize to asking very similar question.)
Let ${\sf CWcpx}$ be the category of CW complexes and let ${\sf ...
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92
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Topological space modeled by special topological structures
Let $X$ be a topological space. Suppose it is "modeled by" topological spaces of the form $\text{Spec}(A)$ for some commutative ring $A$, then, (along with some other conditions/structure), we call $...
- 6,023
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166
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Subspaces of compact spaces and quotients of Hausdorff spaces
Let $\operatorname{Top}$ be the class of topological spaces. Furthermore, let $\mathcal{U}\subset\operatorname{Top}$ and $\mathcal{V}\subset\operatorname{Top}$ classes satisfying the following ...
- 353
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250
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topological properties of $G_{\delta}$ sets in a compact Hausdorff space
I am trying to understand a family of types $\mathcal{F}$ in the set $S(A)$,the set of complete types over $A$ (in the sense of types in model theory) which is a compact and Haurdorff space equipped ...
- 195
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254
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When is the weak topology generated by a family of functions Baire?
Suppose we are given a locally compact space $X$ with $C_b(X)$ denoting the continuous bounded complex or real functions on $X$.
Now, if $A\subset C_b(X)$ is given, I am trying to figure out when the ...
- 173
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0
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63
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What are the various kinds of graphs that can be defined on $C(X)$
I was considering the space $C(X)$ where $X$ is a topological space and $C(X)$ is the set of all continuous functions from $X$ to $\Bbb R$.
What are the various kinds of graphs that can be defined on ...
- 135
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323
views
Closed and discrete sets
Let $\kappa$ in an uncountable regular cardinal and $X$ be a space and $e(x)=\kappa$, where the ``extent'' $e(X)$ of $X$ is the supremum of the cardinalities of
closed discrete subsets of $X$. My ...
- 63
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0
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44
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Decomposition with given closures [closed]
Let $(X,\mathcal T)$ be a topological space. About the subsets $A,B,C$ of $X$ it is known that $$\mathrm {cl} (C)= A \cup B\,, \quad \mathrm {cl} (A) = A\,, \quad \mathrm {cl} (B) = B\,.$$
Does it ...
- 218
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0
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69
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How exactly to adapt Brown's collapse from monoids to algebras?
In The Geometry of Rewriting Systems (Springerlink behind paywall), Kenneth Brown describes a method to collapse the bar resolution of a monoid. Roughly:
Given a simplicial set $X$ equipped with a ...
1
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0
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66
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Quantification over Nets
On a topological space $X$, a net is defined as a map $A \ni \alpha \longmapsto x_{\alpha} \in X$ from a directed set $A$.
With this, compactness of $X$ (for instance) is equivalent to "every net $(...
- 145
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0
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202
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What is the normalized complex of a simplicial set with a monoid action?
This question is a follow up to this question I posted on Math.SE. I will make this question self-contained, though.
In a certain point on the paper The Geometry of Rewriting Systems, Kenneth Brown ...
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132
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Is the Upper Banach density always zero with respect to some sequence of Finite subset
The following question came to me while reading the paper 'Density in Arbitrary Semigroups' by Hindman and Strauss.
Question: Given an infinite subset $A$ of $\mathbb{N}$ such that $A^c$ is also ...
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