All Questions
1,339 questions with no upvoted or accepted answers
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129
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Single theorem for hybrid of winding number and rotation number?
I am trying to make mathematical sense of some observations from my physics research, so I hope that you will bear with me.
For a complex-valued function $z(t)$ dependent on parameter $t$, I calculate ...
- 121
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answers
74
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Is the reversibility of inflation of a subset equivalent to its smoothness?
$D_r(x)$ denotes a closed ball of radius $r$ centered at $x$.
Definition. Let $M \subset \mathbb{R}^n$.
$D_r (M): = \bigcup\limits_{x \in M} D_r (x)$
$Int_r (M): = \{x ~|~ D_r(x) \subset M\}$
...
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61
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Does absolute retract imply convex structure?
In the theory of selection, it is known that any compact absolute retract (AR) carries a convexity structure defined by E. Michael. It is also known that a convex structure
developed by Van de Vel ...
- 2,083
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58
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A generalization of metrics taking values in partial orders
I'm investigating the origin of the following notion:
Let $S=(S, +, <, 0)$ be a partially ordered semigroup with minimum $0$ (such that $<$ is invariant by the action of $+$ on both sides).
A $S$...
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201
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are acyclic fibrations of nice spaces absolute extensors for perfectly normal spaces?
A space $Y$ is called an absolute extensor for normal spaces (also sometimes solid) if, for any normal space $X$, closed subset $A$ of $X$, and map $f:A\to Y$, there exists a map $f′:X\to Y$ such that ...
- 317
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73
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Nonzero idempotents in compact semitopological semigroups with zero
Let $S$ be a compact semitopological semigroup. Then, $S$ contains minimal idempotents by Ellis' theorem.
Ellis' Theorem: In a compact left-topological (resp. right-topological) semigroup, every ...
- 2,605
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145
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Semigroup ideals of a ring or an algebra
Let $R$ be a ring or an algebra. Suppose $B\subseteq R$ satisfies the property $BR \subseteq B$ and $RB\subseteq B$. Is there a general theory of subsets with this property in a ring (resp. algebra) ...
- 2,605
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84
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On the (Brouwer-Koebe) Continuity Method
The so-called Continuity Method is a simple yet powerful method to show that a given continuous injective map is surjective. Namely, suppose that $f:X \to Y$ is a map between two connected manifolds $...
- 2,154
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119
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The "matrix direct sum" monoid modulo unitary equivalence
Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the ...
- 4,943
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125
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(Local) simple connectedness of irreducible algebraic varieties
Let $\mathbb k$ be an algebraically closed field of characteristic zero.
I have two questions:
(1) Is an irreducible algebraic variety $X/\mathbb k$ of dimension at least 2 locally simply connected?
(...
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2
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222
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Complete reducibility, in linear algebra and in topology
I thought that this is a simple question and asked it at the Mathematics StackExchange, but I now have to elevate it to MathOverflow.
Consider a representation $A(G)$ of a group $G$ in a vector space $...
- 603
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203
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Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?
It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G.
More generally here (MO275769) Qiaochu Yuan ...
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2
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86
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Existence of a proper Morse function
I started with a finite CW complex $X$ of dimension $n\geq 3$. I have embedded (local embedding) it inside $R^{2n}$. Now take a regular neighbourhood $U$ of $X$ in $R^{2n}$ which has the same homotopy ...
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67
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A polar open set in a topological subspace?
Suppose $U$ is a bounded open set in $\mathbb{R}^m$ with ($m\geq2$). Is it possible to have a non-empty set $E$ in the boundary $ \partial U$ of $U$ that is open in $ \partial U$ and is polar?
A set $...
- 411
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165
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Dimension of Cartesian products
Is there a notion of dimension such that for all Borel sets $A,B\subseteq\mathbb{R}^{n}$ we have
$$ \dim(A\times B)=\dim(A)+\dim(B)?$$ For topological, Minkowski, packing and Hausdorff dimension this ...
- 1,648
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162
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Banach–Mazur game and mappings
The Banach-Mazur game on a nonempty space $X$ is defined as follows: two players, $I$ and $II$, alternately choose nonempty open sets
\begin{matrix}
I & U_0 && U_1 && \cdots ...
- 115
2
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answers
74
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Terminology and notation for generated subgroups
I would like to think about formation of the smallest subgroup (or monoid, or whatever) $H$ of $G$ containing two given subgroups $A$ and $B$ as an operation on subgroups, and I wonder if there is a ...
- 12.5k
2
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129
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When every connected component is path connected
Let $X$ be a compact $T_0$ topological space whose connected components are path-connected. Is there any characterization for such a space?
- 71
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83
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Increasing a nowhere dense set in $\mathfrak E_{\mathrm{c}}$
Let $X$ be a closed nowhere dense subset of the complete Erdos space $$\mathfrak E_{\mathrm{c}}=\{x\in \ell^2:x_n\notin \mathbb Q\text{ for all }n<\omega\}.$$
Can you always find a closed nowhere ...
- 3,317
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141
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Example of compact fiber bundle with noncompact fibers
This is a cross post of MSE post somehow:
Is there any example of compact fiber bundle $E$ with noncompact fibers $F$?
Obviously if the base space $B$ is $T_1$ then there is no such example.
- 4,195
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55
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Where can I find literature regarding cardinal invariants of a function space $C(X, Y)$ endowed with the Uniform or Fine topology?
I am working on Function Spaces as a topological space. I want to get a sample paper which studies the cardinal invariants on the function space $C(X, Y)$ rather than on $C(X)$.
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241
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Meagre sets with measure zero
It is well known that meagre sets (in topology theory) and sets of measure zero (in measure theory) are generally not the same things ([O], see also a related question on MO). A set that is small in ...
- 41
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190
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What is the smallest number of nowhere dense affine subsets covering a topological group?
$\DeclareMathOperator\cov{cov}\newcommand\A{\text A}$A subset $A$ of a group $G$ is called affine if $A=xHy$ for some subgroup $H\subseteq G$ and some $x,y\in G$.
Given a non-discrete topological ...
- 42k
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408
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Complete topological groups in which all subgroups are closed
My previous question has been answered by YCor; so I am asking a new one with a reasonable additional assumption. See the previous question for the background and motivation.
General question: does ...
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2
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323
views
Continuous injective functions with dense image
Let $X$ be the set of continuous, injective functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ with dense image; and equip $X$ with the (relative) compact-open topology. What is known about this space?
...
- 5,405
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295
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What topological spaces can be realized as cell complexes?
What are the topological spaces can be realized as cell complexes, up to homeomorphism? It seems for instance that all manifolds can be built from cell complexes. It is clear however that one can ...
- 504
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214
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Products of cones and cones of joins
The join of $A$ and $B$ is the pushout of the diagram
$$
CA \times B \gets A\times B \to A\times CB,
$$
which can be formulated in either the pointed or unpointed topological
category. This pushout is ...
- 12.5k
2
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answers
156
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Topology of the set of polynomials with bounded real algebraic varieties (inside the v. s. of polynomials in $n$ variables and up to degree $d$)
Set $x=(x_{1}, \dots, x_{n}).$ Consider the set $\mathbb{R}[x]_{d}$ of polynomials with coef. in $\mathbb{R}$ in $n$ variables up to degree $d.$ This set can be seen as a finite-dimensional vector ...
- 573
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89
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Semigroups associated to binary necklaces and their semigroup algebra
I came across the following semi-group and the associated finite dimensional semi-group algebras over a field $K$ (which are Nakayama algebras) as they have very nice homological properties. My ...
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2
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70
views
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 ...
- 517
2
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answers
77
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Dense embeddings into Euclidean space
The question is a follow-up on this old post. Fix a positive integer $d$ and consider $\mathbb{R}^d$ with its usual Euclidean topology. Given a metric space $(X,\delta_X)$, what conditions are ...
- 5,405
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49
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Going from a class of path functions to a topology
Someone asked a version of this 10 years ago, but no satisfactory answer was given, so I want to try again.
I have a class of functions F from the reals to a set S (the specifics are complicated and ...
- 459
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answers
96
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Projective objects for compact po-spaces
Let us consider the following definition: a compact po-space is a pair $(X,\leq)$ where $X$ is a compact space and $\leq$ is an order, closed on $X^2$. Then, we can consider the category $KPoSp$ whose ...
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89
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Binary and n-ary topological spaces
I am interested in various generalizations of the notion of topological space; also in topologies placed in untypical frameworks, i.e. intuitionistic topological spaces or nano topological spaces. ...
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96
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Non-commutative version of the order dimension of a poset
I view the order dimension of a poset $P$ as an inherently commutative notion. On the one hand, it can be defined via realizers, which I find fairly intuitive from an order-theoretic viewpoint. On the ...
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148
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Is a closed connected semilattice of $C(I)$ path-connected?
Let $\Gamma $ be a sub-lattice of the Banach space $\big( B(S),\|\cdot\|_\infty\big)$ of all bounded real valued functions on the set $S$ (meaning that for any $f,g\in\Gamma $ both functions $f\wedge ...
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60
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Is the set $\operatorname{Unif}(0,\frac{1}{n})$ for odd and even $n$ a 2-alternating capacity?
Let $\Omega$ be a complete metrizable space $\mathscr A$ its Borel $\sigma$-algebra and $\mathscr M$ the set of all probability measures on $\Omega.$ Every non-empty subset $\mathscr P \subset \...
- 359
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208
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Retracting to a bigger compact
Consider the topological spaces $X$ with the following property:
For every compact $K\subseteq X$ there is a compact set $L$ such that $K\subseteq L\subseteq X$ and $L$ is a retract of $X$.
Let ...
- 128k
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115
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Borel measurability
Suppose we have two locally compact Hausdorff spaces $X$ and $Y$. Let $i:X\to Y$ be a continuous injection. Under what condition the Borel $\sigma$-algebra of $X$ and $i(X)$ are isomorphic via the map ...
- 1,879
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123
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Diffeomorphisms of a "matrix type"
Let $\exp$ denote the matrix exponential map, let $Y\subset C^{\infty}(\mathbb{R}^d,\mathbb{R}^d)$ be defined the collection of all functions of the form
$$
f(x) = \exp\left(
\sum_{i=1}^n f_i(x) A_i
\...
- 5,405
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answers
134
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Stone–Weierstrass theorem for stronger topologies
The Stone–Weierstrass theorem gives an easy to check criterion on a (algebra) set of functions $D\subseteq C(X)$ which ensures that $D$ is dense.
Are any similar results for density in $C_b(X)$ ...
- 5,405
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answers
42
views
Is each zero-dimensional cometrizable space zero-cometrizable?
A topological space $X$ is defined to be
$\bullet$ cometrizable if $X$ admits a weaker metrizable topology $\tau$ such that for every point $x\in X$ and a neighborhood $U_x\subseteq X$ of $x$ there ...
- 42k
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116
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Bijection between monoid of nilpotent decreasing self-maps and local subsemigroup
Let $\mathcal{C}_X\cong\mathcal{C}_n$ be the monoid of self-maps $\alpha$ of $X=\{1\dots,n\}$ that are order-preserving ($\forall x,y$, $x\le y$ $\Rightarrow$ $\alpha(x)\le\alpha(y)$) and decreasing ($...
- 355
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33
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On the number of connected functional digraphs recoverable from the preimage set size structure
I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$.
For example,
$P_j=\left[f^{-j}(...
- 23
2
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0
answers
83
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Holomorphic map proper after shrinking (Kollar's Lecture on Resolution of Singularities)
I'm reading Janos Kolloar's Lecture on Resolution of Singularities and have some problems to understand a detail in the proof of Thm. 1.5 on page 10:
Thm 1.5 (Riemann) Let $F(x,y)$ be an ...
- 6,038
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answers
91
views
Is the natural action of the monoid of endomorphisms is a complete invariant for group?
Let $\alpha$ and $\beta$ be actions of semigroups $A$ and $B$ on sets $X$ and $Y$ respectively. Recall that $\alpha$ and $\beta$ are called isomorphic if there exists an isomorphism $\phi$ between ...
- 6,962
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115
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Direct limits of compact surfaces with uniformly bounded topology
Suppose we have a directed system of inclusions of compact surfaces with boundary
$$S_1 \hookrightarrow S_2 \hookrightarrow S_3 \cdots $$
such that all of the surfaces $\{S_k\}$ have uniformly bounded ...
- 1,601
2
votes
0
answers
65
views
Projective dimension of the functions with compact support
Let $X$ be a locally compact Hausdorff space. And $C(X)$ the ring of all continous real-valued functions and $J(X)$ the ideal of such functions with compact support.
It is known that $X$ is ...
- 26.5k
2
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2k
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On weak compactness of the unit ball in a reflexive Banach space
It is a well known result in functional analysis that a Banach space $X$ is reflexive if and only if the unit ball is weakly compact (compact in the weak topology). This result is also known as ...
- 364
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263
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Are these two definitions of smooth $k$-manifold as a Euclidean subset equivalent?
I am struggling to reconcile the two definitions of smooth k-manifold in $R^n$ from M.Spivaks Calculus on Manifolds (pg 109) and J.W Minor's Topology from differential point of view (pg 01).
Milnor's ...
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