All Questions
1,340 questions with no upvoted or accepted answers
3
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209
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Nowhere Baire spaces
Studying the article "Barely Baire spaces" of W. Fleissner and K. Kunen, using stationary sets, they show an example of a Baire space whose square is nowhere Baire (we call a space $X$ nowhere Baire ...
- 561
3
votes
0
answers
360
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The product of Lindelöf spaces
Let $X= \prod_{n\in \omega} X_n\subset \prod_{n\in \omega}\aleph_n$, where $X_n \subset \aleph_n$ (where $\aleph_n$ is the space with order topology) is Lindelöf for each $n\in \omega$. My question is ...
- 63
3
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197
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Topological properties of Noetherian affine schemes that do not hold for general Noetherian spectral spaces
I used to think that the only reason why an affine scheme with a Noetherian space can fail to be Noetherian is nilpotents. It turns out that this is not true.
This leads me to the following question: ...
user139949
3
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92
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Is there a T3½ category analogue of the density topology?
Motivation: I understand that various attempts have been made at defining a topology on $\mathbb{R}$ that is an analogue of the density topology ([1]) but for category (and meager sets) instead of ...
- 32.5k
3
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406
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A user guide to the theory on Corks
I am trying to digest the meanings of the corks from the both:
algebraic topology
and
geometry topology
perspectives.
Studying corks is important for understanding the exotic phenomenon of 4-...
- 10.5k
3
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0
answers
64
views
Metrically homogeneous spaces as inverse limits
Let $(X,d)$ be a locally compact, separable, connected and $\sigma$-compact metric space such that the group of isometries $G$ acts transitively on $X$. The question is the following:
Is $X$ ...
- 541
3
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78
views
Name for mappings that are "not quite projections"
Is there a known name for the following definition?
Consider topological spaces $X$, $Y$ and $f: X \rightarrow Y$ a continuous mapping. Then, $f$ is an "almost projection" if there is a topological ...
- 1,368
3
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107
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Slightly finer topology on a connected space
Let $(X,\tau)$ be a connected Hausdorff space.
Suppose $S\subseteq X$ is such that for every $U\in\tau$, $$U\cap S\neq\varnothing \implies U\cap \overline S\setminus S\neq\varnothing.$$
Is it ...
- 107
3
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108
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Topological characterization of closed surfaces
I am looking some reference, if it exist, that generalized the Moore`s characterization of the 2-sphere. To be more precise, Moore characterized 2-sphere by these two axioms: A space X is a 2-sphere ...
- 131
3
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209
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Compactification of Tychonoff spaces without full axiom of choice
If $X$ is a Tychonoff space, then using the Tychonoff theorem and thus the full axiom of choice, it follows that $X$ admits a Hausdorff compactification.
My question is : what remains true if we do ...
- 506
3
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102
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Find a certain triangulation subordinate to a given covering of a manifold
Let $\{U_\alpha\}$ be a covering of a smooth manifold $M$. Replacing it by a refined covering if necessary, we may assume some good properties of it, like, (1) any intersection $\cap_{i=1}^k U_{\...
- 2,789
3
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83
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Is $\Box_{n\in\omega} \mathbb{R}$ metacompact?
Is $\Box_{n\in\omega} \mathbb{R}$ (that is $\mathbb{R}^\omega$ endowed with the box topology) metacompact?
- 48.9k
3
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208
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Identification of ultrafilters with measures
We know that each ultrafilter $p$ on $\mathbb{N}$ can be identified with a finitely additive $\{0,1\}$-valued probablity measure $\mu_{p}$ on the power set of $\mathbb{N}$.
Now my question is which ...
- 883
3
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650
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description of dual space of space of Radon measure equipped with topology of weak convergence
Let $\mathcal{M}(\mathbb R)$ be the space of Radon measures, equipped with topology $\tau$ generated by the following "weak convergence":
$$
\mu_n \rightarrow \mu \quad \text{iff} \quad \int f d\mu_n ...
- 325
3
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88
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Which spaces are still Lindelöf after forcing with a Suslin tree?
Let $T$ be a Suslin tree and $f:T\to Y$ be continuous. ($T$ is endowed with the order topology.) Assume that the image of $T$ is contained in a Lindelöf subset of $Y$. Then, force with $T$. Which ...
- 1,855
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80
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On the compactification of partial semigroups
We begin by introducing some relevant definitions.
Definition: A $\textit{partial semigroup}$ is a pair $(S,.)$ where $.$ maps a subset of $S \times S$ to $S$ and for all $a,b,c \in S, (a.b).c=a.(b.c)...
- 745
3
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147
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One strong fixed-point property
Each topological space $A$ with fixed-point property is connected (all clopen subsets are trivial). This is an analog of Rice theorem (all decidable subsets are trivial). Suppose, we have a space $A$ ...
3
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answers
141
views
Is there a normal space with a $G_\delta$ diagonal which is not submetrizable?
A space has a $G_\delta$-diagonal if its diagonal can be written as the intersection of countably many open subsets of the square. A space is submetrizable if it has a weaker metrizable topology. ...
- 4,413
3
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answers
154
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$G_\delta$-diagonal and productivity of the CCC
Is there a known example of a completely regular c.c.c. space with $G_\delta$-diagonal which is not productively c.c.c.?
The non-existence of such a space is consistent (for example, under $MA$ no ...
- 1,615
3
votes
0
answers
78
views
Is every weakly Lindelof Banach space a $D$-space?
An open neighbourhood assignment for a topological space $(X, \tau)$ is a map $U: X \to \tau$ such that $x \in U(x)$, for every $x \in X$. A space $X$ is called a $D$-space if for every open ...
- 4,413
3
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92
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Is the elasticity of a submonoid of the free abelian monoid over a finite set either rational or infinite?
Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$, and $H$ a submonoid of $\mathscr F(P)$.
Given $x \in H \setminus \{1_H\}$, we let $\mathsf L_H(x)$ be the set of all $...
- 10.5k
3
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answers
56
views
Name for a special kind of neighborhood assignment or for the existence thereof
Lets say temporarily that a topological space $(X,\tau)$ is weird if there is a function $\varphi:X \to \tau$ such that for all $x \in X$:
$x\in\varphi(x)$,
$\{y\in X: x \in \varphi(y)\}$ is finite.
...
- 11.5k
3
votes
0
answers
82
views
Proving the existence of a continuous function that satisfy a certain property from a finite version of this property
Let $M \subseteq [0,1] \times \mathbb{R}^n$ be a compact semialgebraic set.
In particular, $M$ can be described by a finite set of polynomial equalities and inequalities.
Let $\delta_0 > 0$ be a ...
- 745
3
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161
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A characterization of Cauchy filters on countable metric spaces?
Given a filter $\mathcal F$ on a countable set $X$, consider the family
$$\mathcal F^+:=\{A\subset X:\forall F\in\mathcal F\;(A\cap F\ne\emptyset)\}.$$
The following characterization is well-known.
...
- 42k
3
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0
answers
79
views
Semigroups containing an ideal with a local identity
I'm looking for some classes of semigroups containing a (proper) ideal with a local identity (i.e., ideal submonoid). Can somebody give some examples or/and theorems for the followings cases:
(a) ...
- 995
3
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answers
137
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Can monoids of "continuous words" be realized as initial monoid objects?
Whenever $X$ is a set, write $X^*$ for the monoid freely generated by $X$. The elements of $X$ are, of course, words in the letters $X$. When $X$ is finite, there also seems to be a great many ...
- 3,793
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110
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Functorial description of irreducibility of topological space?
This is a crosspost of this MSE question.
A topological space is connected if it's not the coproduct of two non-trivial spaces. Equivalently, it is connected if the copresheaf it represents preserves ...
- 10.5k
3
votes
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answers
118
views
Weak contractibility of some infinite dimensional metric spaces
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that:
$X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite type$^4$, ...
3
votes
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answers
78
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Nowhere dense covering number of a connected $T_2$ space
This is a generalization of an older question.
If $(X,\tau)$ is a connected $T_2$ space with more than 1 point, we define its nowhere dense covering number $\nu(X)$ by the smallest cardinality that a ...
- 48.9k
3
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answers
47
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Counting the monic atoms $f$ in the semiring $\mathbf N[x]$ with $f(0)=1$, bounded coefficients, and degree $k$ (in the limit as $k \to \infty$)
Let $H$ be the multiplicative monoid of the (usual) semiring of polynomials in one variable $x$ with coefficients in $\mathbf N$. Given $\alpha, k \in \mathbf N$, denote by $\mathcal A_k(\alpha)$ the ...
- 10.5k
3
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answers
131
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quantitative winding number?
In light of Tao's discussion of Quantitative Continuity I've decided to post some thoughts on Winding number.
The Jordan curve theorem is non-trivial because closed curves can be very complicated. (...
- 22.8k
3
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answers
143
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Is an Abelian topological group compact if it is complete and Bohr-compact?
A topological group $G$ will be called Bohr-compact if its Bohr topology (i.e., the largest precompact group topology) is compact and Hausdorff.
A topological group $G$ is Bohr-compact if it admits ...
- 42k
3
votes
0
answers
92
views
Arithmetic progressions inside non meager sets
If $A \subseteq \mathbb{R}$ is non-meager Borel set, then $A$ contains arithmetic progressions of every finite length. I know that this is false if we do not assume that $A$ is Borel. In particular, ...
- 31
3
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0
answers
156
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Topology of the Hamel basis in a TVS
Let $V$ be a complex topological vector space, and let $I$ be a Hamel basis of it. Then as a subset $I\subset V$ acquires an induced topology, becoming a topological space. For a topological space $X$ ...
- 1,959
3
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0
answers
125
views
Commutative discrete cyclic operator groups on topological vector spaces
Let $V$ be a complex Hausdorff separable topological vector space of infinite dimensions. Does there exist a commutative discrete subgroup $A\subset\mathcal{L}(V)$ of continuous operators on $V$ with ...
- 1,959
3
votes
0
answers
207
views
Is the homeomorphism group of a Polish space a measurable group?
Let $X$ be a Polish space. Let $H(X)$ be the set of homeomorphisms $h \colon X \to X$, equipped with the "evaluation $\sigma$-algebra", namely $\sigma(h \mapsto h(x) : x \in X)$.
(Note that for any ...
- 708
3
votes
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110
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Distributive lattices -> left regular bands -> Atomistic lower semimodular lattices
Consider the following construction : let $(L,\vee,\wedge)$ be a finite distributive lattice, and let $(\mathrm{Int}(L),\star)$ be the monoid defined on the set of non empty intervals of $L$
$$\mathrm{...
- 2,751
3
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0
answers
96
views
Given a primitive finite set $A\subseteq\bf N$ with $0\in A$, find two more primitive sets $B,C\subseteq\bf N$ with $B\ne C$ and $A+B=A+C$
Let $\mathcal P_{{\rm fin},0}(\mathbf N)$ be the monoid of all finite subsets of $\mathbf N$ containing $0$ with the operation of set addition
$$
(X, Y) \mapsto X + Y := \{x+y: x \in X \text{ and }y \...
- 10.5k
3
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0
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92
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On the eventually regular monoids and generally regular monoids
Planning the problem:
First we give some definitions.
An element $s \in S$ is called eventually regular if for every $s \in S$ there exist a natural number $n$ in $\mathbb{N}$ and $x
\in S$ such ...
- 601
3
votes
0
answers
359
views
Cubical approximation theorem for cubical complexes
A version of the simplicial approximation theorem states that a continuous map between finite simplicial complexes is homotopic to a simplicial map after subdividing the domain.
I have found a claim ...
- 985
3
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0
answers
83
views
Is the increasing union of disk bundles a disk bundle?
Setup: Let $B$ be a $C^r$ $n$-manifold ($r \geq 1$) and $M$ a closed $k$-dimensional $C^r$ submanifold of $B$. Assume there exists a smooth retraction $p:B \to M$ which is also a submersion, so that $...
- 501
3
votes
0
answers
104
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A link of four 2-tori $T^2$ in $S^2 \times S^2$
Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary with their three $S^1$ boundaries of $T^3$ cyclic permuted to obtain a new 4-...
- 10.5k
3
votes
0
answers
106
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A link of four 2-tori $T^2$ in $S^3 \times S^1 \# S^2 \times S^2 \# S^2 \times S^2$
Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary to obtain a new 4-manifold:
$$(S^4 \smallsetminus D^2\times T^2) \cup (S^4 \...
- 10.5k
3
votes
0
answers
609
views
Homeomorphism Between $C([0,1])$ and $C([0,1]^2)$
It is well known that there cannot be a homeomorphism between $\mathbb{R}$ and $\mathbb{R^2}$ because if we remove the origin from $\mathbb{R}$ the space becomes disconnected, but if we remove the ...
- 371
3
votes
0
answers
132
views
Duality for continuous lattices based on [0, 1]
A continuous lattice may be defined as a complete lattice in which arbitrary meets distribute over directed joins. A continuous lattice is naturally regarded as an algebraic structure where the ...
- 133
3
votes
0
answers
103
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Separation-free topological completeness notion
Cannot really claim that I have immediate urgent motivation to study this question but it appeared to me long ago, I recalled it now by some reason and decided to ask it here.
There is a strong ...
- 18.4k
3
votes
0
answers
182
views
LCH topologies on Groups that are not group topologies
Ellis's 1957 paper on Locally Compact Transformation groups proves the following:
A locally compact hausdorff topology on a group $(G, \cdot)$ for which left and right multiplication are (separately)...
- 352
3
votes
0
answers
198
views
Properties of convergence at points of continuity
Let $J$ denote the set of functions $f : [0, \infty) \to \mathbb{R}$ that are right-continuous and have left-hand limits (r.c.l.l.) and such that their points of discontinuity are jumps.
Then $J$ is a ...
- 1,773
3
votes
0
answers
93
views
When closed subsets have finitely many connected componenets
Let $X$ be topological space such that every its closed subset has finitely many connected componenets. Is there any charactrization for such topological space?
- 31
3
votes
0
answers
1k
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Notion of convergence on a dense subset
My motivation for this question is as follows.
Consider the set $D$ of cadlag functions on $(0,1)$ and its subset $D^\uparrow$ of non-decreasing cadlag functions.
Each $f \in D$ has at most countably ...
- 1,773